inifcns_nstdsums.cpp

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/*
 *  GiNaC Copyright (C) 1999-2011 Johannes Gutenberg University Mainz, Germany
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, write to the Free Software
 *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
 */

#include "inifcns.h"

#include "add.h"
#include "constant.h"
#include "lst.h"
#include "mul.h"
#include "numeric.h"
#include "operators.h"
#include "power.h"
#include "pseries.h"
#include "relational.h"
#include "symbol.h"
#include "utils.h"
#include "wildcard.h"

#include <cln/cln.h>
#include <sstream>
#include <stdexcept>
#include <vector>

namespace GiNaC {


//
// Classical polylogarithm  Li(n,x)
//
// helper functions
//


// anonymous namespace for helper functions
namespace {


// lookup table for factors built from Bernoulli numbers
// see fill_Xn()
std::vector<std::vector<cln::cl_N> > Xn;
// initial size of Xn that should suffice for 32bit machines (must be even)
const int xninitsizestep = 26;
int xninitsize = xninitsizestep;
int xnsize = 0;


// This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
// With these numbers the polylogs can be calculated as follows:
//   Li_p (x)  =  \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with  u = -log(1-x)
//   X_0(n) = B_n (Bernoulli numbers)
//   X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
// The calculation of Xn depends on X0 and X{n-1}.
// X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
// This results in a slightly more complicated algorithm for the X_n.
// The first index in Xn corresponds to the index of the polylog minus 2.
// The second index in Xn corresponds to the index from the actual sum.
void fill_Xn(int n)
{
    if (n>1) {
        // calculate X_2 and higher (corresponding to Li_4 and higher)
        std::vector<cln::cl_N> buf(xninitsize);
        std::vector<cln::cl_N>::iterator it = buf.begin();
        cln::cl_N result;
        *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
        it++;
        for (int i=2; i<=xninitsize; i++) {
            if (i&1) {
                result = 0; // k == 0
            } else {
                result = Xn[0][i/2-1]; // k == 0
            }
            for (int k=1; k<i-1; k++) {
                if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
                    result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
                }
            }
            result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
            result = result + Xn[n-1][i-1] / (i+1); // k == i
            
            *it = result;
            it++;
        }
        Xn.push_back(buf);
    } else if (n==1) {
        // special case to handle the X_0 correct
        std::vector<cln::cl_N> buf(xninitsize);
        std::vector<cln::cl_N>::iterator it = buf.begin();
        cln::cl_N result;
        *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
        it++;
        *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
        it++;
        for (int i=3; i<=xninitsize; i++) {
            if (i & 1) {
                result = -Xn[0][(i-3)/2]/2;
                *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
                it++;
            } else {
                result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
                for (int k=1; k<i/2; k++) {
                    result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
                }
                *it = result;
                it++;
            }
        }
        Xn.push_back(buf);
    } else {
        // calculate X_0
        std::vector<cln::cl_N> buf(xninitsize/2);
        std::vector<cln::cl_N>::iterator it = buf.begin();
        for (int i=1; i<=xninitsize/2; i++) {
            *it = bernoulli(i*2).to_cl_N();
            it++;
        }
        Xn.push_back(buf);
    }

    xnsize++;
}

// doubles the number of entries in each Xn[]
void double_Xn()
{
    const int pos0 = xninitsize / 2;
    // X_0
    for (int i=1; i<=xninitsizestep/2; ++i) {
        Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
    }
    if (Xn.size() > 1) {
        int xend = xninitsize + xninitsizestep;
        cln::cl_N result;
        // X_1
        for (int i=xninitsize+1; i<=xend; ++i) {
            if (i & 1) {
                result = -Xn[0][(i-3)/2]/2;
                Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result);
            } else {
                result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
                for (int k=1; k<i/2; k++) {
                    result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
                }
                Xn[1].push_back(result);
            }
        }
        // X_n
        for (size_t n=2; n<Xn.size(); ++n) {
            for (int i=xninitsize+1; i<=xend; ++i) {
                if (i & 1) {
                    result = 0; // k == 0
                } else {
                    result = Xn[0][i/2-1]; // k == 0
                }
                for (int k=1; k<i-1; ++k) {
                    if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
                        result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
                    }
                }
                result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
                result = result + Xn[n-1][i-1] / (i+1); // k == i
                Xn[n].push_back(result);
            }
        }
    }
    xninitsize += xninitsizestep;
}


// calculates Li(2,x) without Xn
cln::cl_N Li2_do_sum(const cln::cl_N& x)
{
    cln::cl_N res = x;
    cln::cl_N resbuf;
    cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
    cln::cl_I den = 1; // n^2 = 1
    unsigned i = 3;
    do {
        resbuf = res;
        num = num * x;
        den = den + i;  // n^2 = 4, 9, 16, ...
        i += 2;
        res = res + num / den;
    } while (res != resbuf);
    return res;
}


// calculates Li(2,x) with Xn
cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
{
    std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
    std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
    cln::cl_N u = -cln::log(1-x);
    cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
    cln::cl_N uu = cln::square(u);
    cln::cl_N res = u - uu/4;
    cln::cl_N resbuf;
    unsigned i = 1;
    do {
        resbuf = res;
        factor = factor * uu / (2*i * (2*i+1));
        res = res + (*it) * factor;
        i++;
        if (++it == xend) {
            double_Xn();
            it = Xn[0].begin() + (i-1);
            xend = Xn[0].end();
        }
    } while (res != resbuf);
    return res;
}


// calculates Li(n,x), n>2 without Xn
cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
{
    cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
    cln::cl_N res = x;
    cln::cl_N resbuf;
    int i=2;
    do {
        resbuf = res;
        factor = factor * x;
        res = res + factor / cln::expt(cln::cl_I(i),n);
        i++;
    } while (res != resbuf);
    return res;
}


// calculates Li(n,x), n>2 with Xn
cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
{
    std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
    std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
    cln::cl_N u = -cln::log(1-x);
    cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
    cln::cl_N res = u;
    cln::cl_N resbuf;
    unsigned i=2;
    do {
        resbuf = res;
        factor = factor * u / i;
        res = res + (*it) * factor;
        i++;
        if (++it == xend) {
            double_Xn();
            it = Xn[n-2].begin() + (i-2);
            xend = Xn[n-2].end();
        }
    } while (res != resbuf);
    return res;
}


// forward declaration needed by function Li_projection and C below
const cln::cl_N S_num(int n, int p, const cln::cl_N& x);


// helper function for classical polylog Li
cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
{
    // treat n=2 as special case
    if (n == 2) {
        // check if precalculated X0 exists
        if (xnsize == 0) {
            fill_Xn(0);
        }

        if (cln::realpart(x) < 0.5) {
            // choose the faster algorithm
            // the switching point was empirically determined. the optimal point
            // depends on hardware, Digits, ... so an approx value is okay.
            // it solves also the problem with precision due to the u=-log(1-x) transformation
            if (cln::abs(cln::realpart(x)) < 0.25) {
                
                return Li2_do_sum(x);
            } else {
                return Li2_do_sum_Xn(x);
            }
        } else {
            // choose the faster algorithm
            if (cln::abs(cln::realpart(x)) > 0.75) {
                return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
            } else {
                return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
            }
        }
    } else {
        // check if precalculated Xn exist
        if (n > xnsize+1) {
            for (int i=xnsize; i<n-1; i++) {
                fill_Xn(i);
            }
        }

        if (cln::realpart(x) < 0.5) {
            // choose the faster algorithm
            // with n>=12 the "normal" summation always wins against the method with Xn
            if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
                return Lin_do_sum(n, x);
            } else {
                return Lin_do_sum_Xn(n, x);
            }
        } else {
            cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
            for (int j=0; j<n-1; j++) {
                result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
                                  * cln::expt(cln::log(x), j) / cln::factorial(j);
            }
            return result;
        }
    }
}

// helper function for classical polylog Li
const cln::cl_N Lin_numeric(const int n, const cln::cl_N& x)
{
    if (n == 1) {
        // just a log
        return -cln::log(1-x);
    }
    if (zerop(x)) {
        return 0;
    }
    if (x == 1) {
        // [Kol] (2.22)
        return cln::zeta(n);
    }
    else if (x == -1) {
        // [Kol] (2.22)
        return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
    }
    if (cln::abs(realpart(x)) < 0.4 && cln::abs(cln::abs(x)-1) < 0.01) {
        cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
        for (int j=0; j<n-1; j++) {
            result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
                * cln::expt(cln::log(x), j) / cln::factorial(j);
        }
        return result;
    }

    // what is the desired float format?
    // first guess: default format
    cln::float_format_t prec = cln::default_float_format;
    const cln::cl_N value = x;
    // second guess: the argument's format
    if (!instanceof(realpart(x), cln::cl_RA_ring))
        prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
    else if (!instanceof(imagpart(x), cln::cl_RA_ring))
        prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
    
    // [Kol] (5.15)
    if (cln::abs(value) > 1) {
        cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
        // check if argument is complex. if it is real, the new polylog has to be conjugated.
        if (cln::zerop(cln::imagpart(value))) {
            if (n & 1) {
                result = result + conjugate(Li_projection(n, cln::recip(value), prec));
            }
            else {
                result = result - conjugate(Li_projection(n, cln::recip(value), prec));
            }
        }
        else {
            if (n & 1) {
                result = result + Li_projection(n, cln::recip(value), prec);
            }
            else {
                result = result - Li_projection(n, cln::recip(value), prec);
            }
        }
        cln::cl_N add;
        for (int j=0; j<n-1; j++) {
            add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
                        * Lin_numeric(n-j,1) * cln::expt(cln::log(-value),j) / cln::factorial(j);
        }
        result = result - add;
        return result;
    }
    else {
        return Li_projection(n, value, prec);
    }
}


} // end of anonymous namespace


//
// Multiple polylogarithm  Li(n,x)
//
// helper function
//


// anonymous namespace for helper function
namespace {


// performs the actual series summation for multiple polylogarithms
cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
{
    // ensure all x <> 0.
    for (std::vector<cln::cl_N>::const_iterator it = x.begin(); it != x.end(); ++it) {
        if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits));
    }

    const int j = s.size();
    bool flag_accidental_zero = false;

    std::vector<cln::cl_N> t(j);
    cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));

    cln::cl_N t0buf;
    int q = 0;
    do {
        t0buf = t[0];
        q++;
        t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
        for (int k=j-2; k>=0; k--) {
            t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
        }
        q++;
        t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
        for (int k=j-2; k>=0; k--) {
            flag_accidental_zero = cln::zerop(t[k+1]);
            t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
        }
    } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero );

    return t[0];
}


// forward declaration for Li_eval()
lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);


// type used by the transformation functions for G
typedef std::vector<int> Gparameter;


// G_eval1-function for G transformations
ex G_eval1(int a, int scale, const exvector& gsyms)
{
    if (a != 0) {
        const ex& scs = gsyms[std::abs(scale)];
        const ex& as = gsyms[std::abs(a)];
        if (as != scs) {
            return -log(1 - scs/as);
        } else {
            return -zeta(1);
        }
    } else {
        return log(gsyms[std::abs(scale)]);
    }
}


// G_eval-function for G transformations
ex G_eval(const Gparameter& a, int scale, const exvector& gsyms)
{
    // check for properties of G
    ex sc = gsyms[std::abs(scale)];
    lst newa;
    bool all_zero = true;
    bool all_ones = true;
    int count_ones = 0;
    for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
        if (*it != 0) {
            const ex sym = gsyms[std::abs(*it)];
            newa.append(sym);
            all_zero = false;
            if (sym != sc) {
                all_ones = false;
            }
            if (all_ones) {
                ++count_ones;
            }
        } else {
            all_ones = false;
        }
    }

    // care about divergent G: shuffle to separate divergencies that will be canceled
    // later on in the transformation
    if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
        // do shuffle
        Gparameter short_a;
        Gparameter::const_iterator it = a.begin();
        ++it;
        for (; it != a.end(); ++it) {
            short_a.push_back(*it);
        }
        ex result = G_eval1(a.front(), scale, gsyms) * G_eval(short_a, scale, gsyms);
        it = short_a.begin();
        for (int i=1; i<count_ones; ++i) {
            ++it;
        }
        for (; it != short_a.end(); ++it) {

            Gparameter newa;
            Gparameter::const_iterator it2 = short_a.begin();
            for (; it2 != it; ++it2) {
                newa.push_back(*it2);
            }
            newa.push_back(*it);
            newa.push_back(a[0]);
            it2 = it;
            ++it2;
            for (; it2 != short_a.end(); ++it2) {
                newa.push_back(*it2);   
            }
            result -= G_eval(newa, scale, gsyms);
        }
        return result / count_ones;
    }

    // G({1,...,1};y) -> G({1};y)^k / k!
    if (all_ones && a.size() > 1) {
        return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones);
    }

    // G({0,...,0};y) -> log(y)^k / k!
    if (all_zero) {
        return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
    }

    // no special cases anymore -> convert it into Li
    lst m;
    lst x;
    ex argbuf = gsyms[std::abs(scale)];
    ex mval = _ex1;
    for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) {
        if (*it != 0) {
            const ex& sym = gsyms[std::abs(*it)];
            x.append(argbuf / sym);
            m.append(mval);
            mval = _ex1;
            argbuf = sym;
        } else {
            ++mval;
        }
    }
    return pow(-1, x.nops()) * Li(m, x);
}


// converts data for G: pending_integrals -> a
Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
{
    GINAC_ASSERT(pending_integrals.size() != 1);

    if (pending_integrals.size() > 0) {
        // get rid of the first element, which would stand for the new upper limit
        Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
        return new_a;
    } else {
        // just return empty parameter list
        Gparameter new_a;
        return new_a;
    }
}


// check the parameters a and scale for G and return information about convergence, depth, etc.
// convergent     : true if G(a,scale) is convergent
// depth          : depth of G(a,scale)
// trailing_zeros : number of trailing zeros of a
// min_it         : iterator of a pointing on the smallest element in a
Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
        bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
{
    convergent = true;
    depth = 0;
    trailing_zeros = 0;
    min_it = a.end();
    Gparameter::const_iterator lastnonzero = a.end();
    for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
        if (std::abs(*it) > 0) {
            ++depth;
            trailing_zeros = 0;
            lastnonzero = it;
            if (std::abs(*it) < scale) {
                convergent = false;
                if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
                    min_it = it;
                }
            }
        } else {
            ++trailing_zeros;
        }
    }
    if (lastnonzero == a.end())
        return a.end();
    return ++lastnonzero;
}


// add scale to pending_integrals if pending_integrals is empty
Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
{
    GINAC_ASSERT(pending_integrals.size() != 1);

    if (pending_integrals.size() > 0) {
        return pending_integrals;
    } else {
        Gparameter new_pending_integrals;
        new_pending_integrals.push_back(scale);
        return new_pending_integrals;
    }
}


// handles trailing zeroes for an otherwise convergent integral
ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms)
{
    bool convergent;
    int depth, trailing_zeros;
    Gparameter::const_iterator last, dummyit;
    last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);

    GINAC_ASSERT(convergent);

    if ((trailing_zeros > 0) && (depth > 0)) {
        ex result;
        Gparameter new_a(a.begin(), a.end()-1);
        result += G_eval1(0, scale, gsyms) * trailing_zeros_G(new_a, scale, gsyms);
        for (Gparameter::const_iterator it = a.begin(); it != last; ++it) {
            Gparameter new_a(a.begin(), it);
            new_a.push_back(0);
            new_a.insert(new_a.end(), it, a.end()-1);
            result -= trailing_zeros_G(new_a, scale, gsyms);
        }

        return result / trailing_zeros;
    } else {
        return G_eval(a, scale, gsyms);
    }
}


// G transformation [VSW] (57),(58)
ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale, const exvector& gsyms)
{
    // pendint = ( y1, b1, ..., br )
    //       a = ( 0, ..., 0, amin )
    //   scale = y2
    //
    // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
    // where sr replaces amin

    GINAC_ASSERT(a.back() != 0);
    GINAC_ASSERT(a.size() > 0);

    ex result;
    Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
    const int psize = pending_integrals.size();

    // length == 1
    // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})

    if (a.size() == 1) {

      // ln(-y2_{-+})
      result += log(gsyms[ex_to<numeric>(scale).to_int()]);
        if (a.back() > 0) {
            new_pending_integrals.push_back(-scale);
            result += I*Pi;
        } else {
            new_pending_integrals.push_back(scale);
            result -= I*Pi;
        }
        if (psize) {
            result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
                                   pending_integrals.front(),
                           gsyms);
        }
        
        // G(y2_{-+}; sr)
        result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
                               new_pending_integrals.front(),
                       gsyms);
        
        // G(0; sr)
        new_pending_integrals.back() = 0;
        result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
                               new_pending_integrals.front(),
                       gsyms);

        return result;
    }

    // length > 1
    // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
    //                            - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )

    //term zeta_m
    result -= zeta(a.size());
    if (psize) {
        result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
                               pending_integrals.front(),
                       gsyms);
    }
    
    // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
    //    = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
    Gparameter new_a(a.begin()+1, a.end());
    new_pending_integrals.push_back(0);
    result -= depth_one_trafo_G(new_pending_integrals, new_a, scale, gsyms);
    
    // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
    //    = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
    Gparameter new_pending_integrals_2;
    new_pending_integrals_2.push_back(scale);
    new_pending_integrals_2.push_back(0);
    if (psize) {
        result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
                               pending_integrals.front(),
                       gsyms)
                  * depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
    } else {
        result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
    }

    return result;
}


// forward declaration
ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
         const Gparameter& pendint, const Gparameter& a_old, int scale,
         const exvector& gsyms);


// G transformation [VSW]
ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale,
           const exvector& gsyms)
{
    // main recursion routine
    //
    // pendint = ( y1, b1, ..., br )
    //       a = ( a1, ..., amin, ..., aw )
    //   scale = y2
    //
    // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
    // where sr replaces amin

    // find smallest alpha, determine depth and trailing zeros, and check for convergence
    bool convergent;
    int depth, trailing_zeros;
    Gparameter::const_iterator min_it;
    Gparameter::const_iterator firstzero = 
        check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
    int min_it_pos = min_it - a.begin();

    // special case: all a's are zero
    if (depth == 0) {
        ex result;

        if (a.size() == 0) {
          result = 1;
        } else {
          result = G_eval(a, scale, gsyms);
        }
        if (pendint.size() > 0) {
          result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
                                 pendint.front(),
                         gsyms);
        } 
        return result;
    }

    // handle trailing zeros
    if (trailing_zeros > 0) {
        ex result;
        Gparameter new_a(a.begin(), a.end()-1);
        result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms);
        for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) {
            Gparameter new_a(a.begin(), it);
            new_a.push_back(0);
            new_a.insert(new_a.end(), it, a.end()-1);
            result -= G_transform(pendint, new_a, scale, gsyms);
        }
        return result / trailing_zeros;
    }

    // convergence case
    if (convergent) {
        if (pendint.size() > 0) {
            return G_eval(convert_pending_integrals_G(pendint),
                      pendint.front(), gsyms)*
                G_eval(a, scale, gsyms);
        } else {
            return G_eval(a, scale, gsyms);
        }
    }

    // call basic transformation for depth equal one
    if (depth == 1) {
        return depth_one_trafo_G(pendint, a, scale, gsyms);
    }

    // do recursion
    // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
    //  =  int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
    //   + int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) int_0^{sr} ds_{r+1} d/ds_{r+1} G(a1,...,s_{r+1},...,aw,y2)

    // smallest element in last place
    if (min_it + 1 == a.end()) {
        do { --min_it; } while (*min_it == 0);
        Gparameter empty;
        Gparameter a1(a.begin(),min_it+1);
        Gparameter a2(min_it+1,a.end());

        ex result = G_transform(pendint, a2, scale, gsyms)*
            G_transform(empty, a1, scale, gsyms);

        result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms);
        return result;
    }

    Gparameter empty;
    Gparameter::iterator changeit;

    // first term G(a_1,..,0,...,a_w;a_0)
    Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
    Gparameter new_a = a;
    new_a[min_it_pos] = 0;
    ex result = G_transform(empty, new_a, scale, gsyms);
    if (pendint.size() > 0) {
        result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
                               pendint.front(), gsyms);
    }

    // other terms
    changeit = new_a.begin() + min_it_pos;
    changeit = new_a.erase(changeit);
    if (changeit != new_a.begin()) {
        // smallest in the middle
        new_pendint.push_back(*changeit);
        result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
                               new_pendint.front(), gsyms)*
            G_transform(empty, new_a, scale, gsyms);
        int buffer = *changeit;
        *changeit = *min_it;
        result += G_transform(new_pendint, new_a, scale, gsyms);
        *changeit = buffer;
        new_pendint.pop_back();
        --changeit;
        new_pendint.push_back(*changeit);
        result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
                               new_pendint.front(), gsyms)*
            G_transform(empty, new_a, scale, gsyms);
        *changeit = *min_it;
        result -= G_transform(new_pendint, new_a, scale, gsyms);
    } else {
        // smallest at the front
        new_pendint.push_back(scale);
        result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
                               new_pendint.front(), gsyms)*
            G_transform(empty, new_a, scale, gsyms);
        new_pendint.back() =  *changeit;
        result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
                               new_pendint.front(), gsyms)*
            G_transform(empty, new_a, scale, gsyms);
        *changeit = *min_it;
        result += G_transform(new_pendint, new_a, scale, gsyms);
    }
    return result;
}


// shuffles the two parameter list a1 and a2 and calls G_transform for every term except
// for the one that is equal to a_old
ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
         const Gparameter& pendint, const Gparameter& a_old, int scale,
         const exvector& gsyms) 
{
    if (a1.size()==0 && a2.size()==0) {
        // veto the one configuration we don't want
        if ( a0 == a_old ) return 0;

        return G_transform(pendint, a0, scale, gsyms);
    }

    if (a2.size()==0) {
        Gparameter empty;
        Gparameter aa0 = a0;
        aa0.insert(aa0.end(),a1.begin(),a1.end());
        return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms);
    }

    if (a1.size()==0) {
        Gparameter empty;
        Gparameter aa0 = a0;
        aa0.insert(aa0.end(),a2.begin(),a2.end());
        return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms);
    }

    Gparameter a1_removed(a1.begin()+1,a1.end());
    Gparameter a2_removed(a2.begin()+1,a2.end());

    Gparameter a01 = a0;
    Gparameter a02 = a0;

    a01.push_back( a1[0] );
    a02.push_back( a2[0] );

    return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms)
         + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms);
}

// handles the transformations and the numerical evaluation of G
// the parameter x, s and y must only contain numerics
static cln::cl_N
G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
      const cln::cl_N& y);

// do acceleration transformation (hoelder convolution [BBB])
// the parameter x, s and y must only contain numerics
static cln::cl_N
G_do_hoelder(std::vector<cln::cl_N> x, /* yes, it's passed by value */
         const std::vector<int>& s, const cln::cl_N& y)
{
    cln::cl_N result;
    const std::size_t size = x.size();
    for (std::size_t i = 0; i < size; ++i)
        x[i] = x[i]/y;

    for (std::size_t r = 0; r <= size; ++r) {
        cln::cl_N buffer(1 & r ? -1 : 1);
        cln::cl_RA p(2);
        bool adjustp;
        do {
            adjustp = false;
            for (std::size_t i = 0; i < size; ++i) {
                if (x[i] == cln::cl_RA(1)/p) {
                    p = p/2 + cln::cl_RA(3)/2;
                    adjustp = true;
                    continue;
                }
            }
        } while (adjustp);
        cln::cl_RA q = p/(p-1);
        std::vector<cln::cl_N> qlstx;
        std::vector<int> qlsts;
        for (std::size_t j = r; j >= 1; --j) {
            qlstx.push_back(cln::cl_N(1) - x[j-1]);
            if (instanceof(x[j-1], cln::cl_R_ring) &&
                realpart(x[j-1]) > 1 && realpart(x[j-1]) <= 2) {
                qlsts.push_back(s[j-1]);
            } else {
                qlsts.push_back(-s[j-1]);
            }
        }
        if (qlstx.size() > 0) {
            buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
        }
        std::vector<cln::cl_N> plstx;
        std::vector<int> plsts;
        for (std::size_t j = r+1; j <= size; ++j) {
            plstx.push_back(x[j-1]);
            plsts.push_back(s[j-1]);
        }
        if (plstx.size() > 0) {
            buffer = buffer*G_numeric(plstx, plsts, 1/p);
        }
        result = result + buffer;
    }
    return result;
}

// convergence transformation, used for numerical evaluation of G function.
// the parameter x, s and y must only contain numerics
static cln::cl_N
G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
       const cln::cl_N& y)
{
    // sort (|x|<->position) to determine indices
    typedef std::multimap<cln::cl_R, std::size_t> sortmap_t;
    sortmap_t sortmap;
    std::size_t size = 0;
    for (std::size_t i = 0; i < x.size(); ++i) {
        if (!zerop(x[i])) {
            sortmap.insert(std::make_pair(abs(x[i]), i));
            ++size;
        }
    }
    // include upper limit (scale)
    sortmap.insert(std::make_pair(abs(y), x.size()));

    // generate missing dummy-symbols
    int i = 1;
    // holding dummy-symbols for the G/Li transformations
    exvector gsyms;
    gsyms.push_back(symbol("GSYMS_ERROR"));
    cln::cl_N lastentry(0);
    for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
        if (it != sortmap.begin()) {
            if (it->second < x.size()) {
                if (x[it->second] == lastentry) {
                    gsyms.push_back(gsyms.back());
                    continue;
                }
            } else {
                if (y == lastentry) {
                    gsyms.push_back(gsyms.back());
                    continue;
                }
            }
        }
        std::ostringstream os;
        os << "a" << i;
        gsyms.push_back(symbol(os.str()));
        ++i;
        if (it->second < x.size()) {
            lastentry = x[it->second];
        } else {
            lastentry = y;
        }
    }

    // fill position data according to sorted indices and prepare substitution list
    Gparameter a(x.size());
    exmap subslst;
    std::size_t pos = 1;
    int scale = pos;
    for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
        if (it->second < x.size()) {
            if (s[it->second] > 0) {
                a[it->second] = pos;
            } else {
                a[it->second] = -int(pos);
            }
            subslst[gsyms[pos]] = numeric(x[it->second]);
        } else {
            scale = pos;
            subslst[gsyms[pos]] = numeric(y);
        }
        ++pos;
    }

    // do transformation
    Gparameter pendint;
    ex result = G_transform(pendint, a, scale, gsyms);
    // replace dummy symbols with their values
    result = result.eval().expand();
    result = result.subs(subslst).evalf();
    if (!is_a<numeric>(result))
        throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
    
    cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
    return ret;
}

// handles the transformations and the numerical evaluation of G
// the parameter x, s and y must only contain numerics
static cln::cl_N
G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
      const cln::cl_N& y)
{
    // check for convergence and necessary accelerations
    bool need_trafo = false;
    bool need_hoelder = false;
    std::size_t depth = 0;
    for (std::size_t i = 0; i < x.size(); ++i) {
        if (!zerop(x[i])) {
            ++depth;
            const cln::cl_N x_y = abs(x[i]) - y;
            if (instanceof(x_y, cln::cl_R_ring) &&
                realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2)))
                need_trafo = true;

            if (abs(abs(x[i]/y) - 1) < 0.01)
                need_hoelder = true;
        }
    }
    if (zerop(x[x.size() - 1]))
        need_trafo = true;

    if (depth == 1 && x.size() == 2 && !need_trafo)
        return - Li_projection(2, y/x[1], cln::float_format(Digits));
    
    // do acceleration transformation (hoelder convolution [BBB])
    if (need_hoelder)
        return G_do_hoelder(x, s, y);
    
    // convergence transformation
    if (need_trafo)
        return G_do_trafo(x, s, y);

    // do summation
    std::vector<cln::cl_N> newx;
    newx.reserve(x.size());
    std::vector<int> m;
    m.reserve(x.size());
    int mcount = 1;
    int sign = 1;
    cln::cl_N factor = y;
    for (std::size_t i = 0; i < x.size(); ++i) {
        if (zerop(x[i])) {
            ++mcount;
        } else {
            newx.push_back(factor/x[i]);
            factor = x[i];
            m.push_back(mcount);
            mcount = 1;
            sign = -sign;
        }
    }

    return sign*multipleLi_do_sum(m, newx);
}


ex mLi_numeric(const lst& m, const lst& x)
{
    // let G_numeric do the transformation
    std::vector<cln::cl_N> newx;
    newx.reserve(x.nops());
    std::vector<int> s;
    s.reserve(x.nops());
    cln::cl_N factor(1);
    for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
        for (int i = 1; i < *itm; ++i) {
            newx.push_back(cln::cl_N(0));
            s.push_back(1);
        }
        const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
        newx.push_back(factor/xi);
        factor = factor/xi;
        s.push_back(1);
    }
    return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
}


} // end of anonymous namespace


//
// Generalized multiple polylogarithm  G(x, y) and G(x, s, y)
//
// GiNaC function
//


static ex G2_evalf(const ex& x_, const ex& y)
{
    if (!y.info(info_flags::positive)) {
        return G(x_, y).hold();
    }
    lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
    if (x.nops() == 0) {
        return _ex1;
    }
    if (x.op(0) == y) {
        return G(x_, y).hold();
    }
    std::vector<int> s;
    s.reserve(x.nops());
    bool all_zero = true;
    for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
        if (!(*it).info(info_flags::numeric)) {
            return G(x_, y).hold();
        }
        if (*it != _ex0) {
            all_zero = false;
        }
        if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
            s.push_back(-1);
        }
        else {
            s.push_back(1);
        }
    }
    if (all_zero) {
        return pow(log(y), x.nops()) / factorial(x.nops());
    }
    std::vector<cln::cl_N> xv;
    xv.reserve(x.nops());
    for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
        xv.push_back(ex_to<numeric>(*it).to_cl_N());
    cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
    return numeric(result);
}


static ex G2_eval(const ex& x_, const ex& y)
{
    //TODO eval to MZV or H or S or Lin

    if (!y.info(info_flags::positive)) {
        return G(x_, y).hold();
    }
    lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
    if (x.nops() == 0) {
        return _ex1;
    }
    if (x.op(0) == y) {
        return G(x_, y).hold();
    }
    std::vector<int> s;
    s.reserve(x.nops());
    bool all_zero = true;
    bool crational = true;
    for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
        if (!(*it).info(info_flags::numeric)) {
            return G(x_, y).hold();
        }
        if (!(*it).info(info_flags::crational)) {
            crational = false;
        }
        if (*it != _ex0) {
            all_zero = false;
        }
        if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
            s.push_back(-1);
        }
        else {
            s.push_back(+1);
        }
    }
    if (all_zero) {
        return pow(log(y), x.nops()) / factorial(x.nops());
    }
    if (!y.info(info_flags::crational)) {
        crational = false;
    }
    if (crational) {
        return G(x_, y).hold();
    }
    std::vector<cln::cl_N> xv;
    xv.reserve(x.nops());
    for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
        xv.push_back(ex_to<numeric>(*it).to_cl_N());
    cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
    return numeric(result);
}


unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
                                evalf_func(G2_evalf).
                                eval_func(G2_eval).
                                do_not_evalf_params().
                                overloaded(2));
//TODO
//                                derivative_func(G2_deriv).
//                                print_func<print_latex>(G2_print_latex).


static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
{
    if (!y.info(info_flags::positive)) {
        return G(x_, s_, y).hold();
    }
    lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
    lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
    if (x.nops() != s.nops()) {
        return G(x_, s_, y).hold();
    }
    if (x.nops() == 0) {
        return _ex1;
    }
    if (x.op(0) == y) {
        return G(x_, s_, y).hold();
    }
    std::vector<int> sn;
    sn.reserve(s.nops());
    bool all_zero = true;
    for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
        if (!(*itx).info(info_flags::numeric)) {
            return G(x_, y).hold();
        }
        if (!(*its).info(info_flags::real)) {
            return G(x_, y).hold();
        }
        if (*itx != _ex0) {
            all_zero = false;
        }
        if ( ex_to<numeric>(*itx).is_real() ) {
            if ( *its >= 0 ) {
                sn.push_back(1);
            }
            else {
                sn.push_back(-1);
            }
        }
        else {
            if ( ex_to<numeric>(*itx).imag() > 0 ) {
                sn.push_back(1);
            }
            else {
                sn.push_back(-1);
            }
        }
    }
    if (all_zero) {
        return pow(log(y), x.nops()) / factorial(x.nops());
    }
    std::vector<cln::cl_N> xn;
    xn.reserve(x.nops());
    for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
        xn.push_back(ex_to<numeric>(*it).to_cl_N());
    cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
    return numeric(result);
}


static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
{
    //TODO eval to MZV or H or S or Lin

    if (!y.info(info_flags::positive)) {
        return G(x_, s_, y).hold();
    }
    lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
    lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
    if (x.nops() != s.nops()) {
        return G(x_, s_, y).hold();
    }
    if (x.nops() == 0) {
        return _ex1;
    }
    if (x.op(0) == y) {
        return G(x_, s_, y).hold();
    }
    std::vector<int> sn;
    sn.reserve(s.nops());
    bool all_zero = true;
    bool crational = true;
    for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
        if (!(*itx).info(info_flags::numeric)) {
            return G(x_, s_, y).hold();
        }
        if (!(*its).info(info_flags::real)) {
            return G(x_, s_, y).hold();
        }
        if (!(*itx).info(info_flags::crational)) {
            crational = false;
        }
        if (*itx != _ex0) {
            all_zero = false;
        }
        if ( ex_to<numeric>(*itx).is_real() ) {
            if ( *its >= 0 ) {
                sn.push_back(1);
            }
            else {
                sn.push_back(-1);
            }
        }
        else {
            if ( ex_to<numeric>(*itx).imag() > 0 ) {
                sn.push_back(1);
            }
            else {
                sn.push_back(-1);
            }
        }
    }
    if (all_zero) {
        return pow(log(y), x.nops()) / factorial(x.nops());
    }
    if (!y.info(info_flags::crational)) {
        crational = false;
    }
    if (crational) {
        return G(x_, s_, y).hold();
    }
    std::vector<cln::cl_N> xn;
    xn.reserve(x.nops());
    for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
        xn.push_back(ex_to<numeric>(*it).to_cl_N());
    cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
    return numeric(result);
}


unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
                                evalf_func(G3_evalf).
                                eval_func(G3_eval).
                                do_not_evalf_params().
                                overloaded(2));
//TODO
//                                derivative_func(G3_deriv).
//                                print_func<print_latex>(G3_print_latex).


//
// Classical polylogarithm and multiple polylogarithm  Li(m,x)
//
// GiNaC function
//


static ex Li_evalf(const ex& m_, const ex& x_)
{
    // classical polylogs
    if (m_.info(info_flags::posint)) {
        if (x_.info(info_flags::numeric)) {
            int m__ = ex_to<numeric>(m_).to_int();
            const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
            const cln::cl_N result = Lin_numeric(m__, x__);
            return numeric(result);
        } else {
            // try to numerically evaluate second argument
            ex x_val = x_.evalf();
            if (x_val.info(info_flags::numeric)) {
                int m__ = ex_to<numeric>(m_).to_int();
                const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
                const cln::cl_N result = Lin_numeric(m__, x__);
                return numeric(result);
            }
        }
    }
    // multiple polylogs
    if (is_a<lst>(m_) && is_a<lst>(x_)) {

        const lst& m = ex_to<lst>(m_);
        const lst& x = ex_to<lst>(x_);
        if (m.nops() != x.nops()) {
            return Li(m_,x_).hold();
        }
        if (x.nops() == 0) {
            return _ex1;
        }
        if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
            return Li(m_,x_).hold();
        }

        for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
            if (!(*itm).info(info_flags::posint)) {
                return Li(m_, x_).hold();
            }
            if (!(*itx).info(info_flags::numeric)) {
                return Li(m_, x_).hold();
            }
            if (*itx == _ex0) {
                return _ex0;
            }
        }

        return mLi_numeric(m, x);
    }

    return Li(m_,x_).hold();
}


static ex Li_eval(const ex& m_, const ex& x_)
{
    if (is_a<lst>(m_)) {
        if (is_a<lst>(x_)) {
            // multiple polylogs
            const lst& m = ex_to<lst>(m_);
            const lst& x = ex_to<lst>(x_);
            if (m.nops() != x.nops()) {
                return Li(m_,x_).hold();
            }
            if (x.nops() == 0) {
                return _ex1;
            }
            bool is_H = true;
            bool is_zeta = true;
            bool do_evalf = true;
            bool crational = true;
            for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
                if (!(*itm).info(info_flags::posint)) {
                    return Li(m_,x_).hold();
                }
                if ((*itx != _ex1) && (*itx != _ex_1)) {
                    if (itx != x.begin()) {
                        is_H = false;
                    }
                    is_zeta = false;
                }
                if (*itx == _ex0) {
                    return _ex0;
                }
                if (!(*itx).info(info_flags::numeric)) {
                    do_evalf = false;
                }
                if (!(*itx).info(info_flags::crational)) {
                    crational = false;
                }
            }
            if (is_zeta) {
                return zeta(m_,x_);
            }
            if (is_H) {
                ex prefactor;
                lst newm = convert_parameter_Li_to_H(m, x, prefactor);
                return prefactor * H(newm, x[0]);
            }
            if (do_evalf && !crational) {
                return mLi_numeric(m,x);
            }
        }
        return Li(m_, x_).hold();
    } else if (is_a<lst>(x_)) {
        return Li(m_, x_).hold();
    }

    // classical polylogs
    if (x_ == _ex0) {
        return _ex0;
    }
    if (x_ == _ex1) {
        return zeta(m_);
    }
    if (x_ == _ex_1) {
        return (pow(2,1-m_)-1) * zeta(m_);
    }
    if (m_ == _ex1) {
        return -log(1-x_);
    }
    if (m_ == _ex2) {
        if (x_.is_equal(I)) {
            return power(Pi,_ex2)/_ex_48 + Catalan*I;
        }
        if (x_.is_equal(-I)) {
            return power(Pi,_ex2)/_ex_48 - Catalan*I;
        }
    }
    if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
        int m__ = ex_to<numeric>(m_).to_int();
        const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
        const cln::cl_N result = Lin_numeric(m__, x__);
        return numeric(result);
    }

    return Li(m_, x_).hold();
}


static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
{
    if (is_a<lst>(m) || is_a<lst>(x)) {
        // multiple polylog
        epvector seq;
        seq.push_back(expair(Li(m, x), 0));
        return pseries(rel, seq);
    }
    
    // classical polylog
    const ex x_pt = x.subs(rel, subs_options::no_pattern);
    if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
        // First special case: x==0 (derivatives have poles)
        if (x_pt.is_zero()) {
            const symbol s;
            ex ser;
            // manually construct the primitive expansion
            for (int i=1; i<order; ++i)
                ser += pow(s,i) / pow(numeric(i), m);
            // substitute the argument's series expansion
            ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
            // maybe that was terminating, so add a proper order term
            epvector nseq;
            nseq.push_back(expair(Order(_ex1), order));
            ser += pseries(rel, nseq);
            // reexpanding it will collapse the series again
            return ser.series(rel, order);
        }
        // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
        throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
    }
    // all other cases should be safe, by now:
    throw do_taylor();  // caught by function::series()
}


static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param < 2);
    if (deriv_param == 0) {
        return _ex0;
    }
    if (m_.nops() > 1) {
        throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
    }
    ex m;
    if (is_a<lst>(m_)) {
        m = m_.op(0);
    } else {
        m = m_;
    }
    ex x;
    if (is_a<lst>(x_)) {
        x = x_.op(0);
    } else {
        x = x_;
    }
    if (m > 0) {
        return Li(m-1, x) / x;
    } else {
        return 1/(1-x);
    }
}


static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
{
    lst m;
    if (is_a<lst>(m_)) {
        m = ex_to<lst>(m_);
    } else {
        m = lst(m_);
    }
    lst x;
    if (is_a<lst>(x_)) {
        x = ex_to<lst>(x_);
    } else {
        x = lst(x_);
    }
    c.s << "\\mathrm{Li}_{";
    lst::const_iterator itm = m.begin();
    (*itm).print(c);
    itm++;
    for (; itm != m.end(); itm++) {
        c.s << ",";
        (*itm).print(c);
    }
    c.s << "}(";
    lst::const_iterator itx = x.begin();
    (*itx).print(c);
    itx++;
    for (; itx != x.end(); itx++) {
        c.s << ",";
        (*itx).print(c);
    }
    c.s << ")";
}


REGISTER_FUNCTION(Li,
                  evalf_func(Li_evalf).
                  eval_func(Li_eval).
                  series_func(Li_series).
                  derivative_func(Li_deriv).
                  print_func<print_latex>(Li_print_latex).
                  do_not_evalf_params());


//
// Nielsen's generalized polylogarithm  S(n,p,x)
//
// helper functions
//


// anonymous namespace for helper functions
namespace {


// lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
// see fill_Yn()
std::vector<std::vector<cln::cl_N> > Yn;
int ynsize = 0; // number of Yn[]
int ynlength = 100; // initial length of all Yn[i]


// This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
// The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
// representing S_{n,p}(x).
// The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
// equivalent Z-sum.
// The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
// representing S_{n,p}(x).
// The calculation of Y_n uses the values from Y_{n-1}.
void fill_Yn(int n, const cln::float_format_t& prec)
{
    const int initsize = ynlength;
    //const int initsize = initsize_Yn;
    cln::cl_N one = cln::cl_float(1, prec);

    if (n) {
        std::vector<cln::cl_N> buf(initsize);
        std::vector<cln::cl_N>::iterator it = buf.begin();
        std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
        *it = (*itprev) / cln::cl_N(n+1) * one;
        it++;
        itprev++;
        // sums with an index smaller than the depth are zero and need not to be calculated.
        // calculation starts with depth, which is n+2)
        for (int i=n+2; i<=initsize+n; i++) {
            *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
            it++;
            itprev++;
        }
        Yn.push_back(buf);
    } else {
        std::vector<cln::cl_N> buf(initsize);
        std::vector<cln::cl_N>::iterator it = buf.begin();
        *it = 1 * one;
        it++;
        for (int i=2; i<=initsize; i++) {
            *it = *(it-1) + 1 / cln::cl_N(i) * one;
            it++;
        }
        Yn.push_back(buf);
    }
    ynsize++;
}


// make Yn longer ... 
void make_Yn_longer(int newsize, const cln::float_format_t& prec)
{

    cln::cl_N one = cln::cl_float(1, prec);

    Yn[0].resize(newsize);
    std::vector<cln::cl_N>::iterator it = Yn[0].begin();
    it += ynlength;
    for (int i=ynlength+1; i<=newsize; i++) {
        *it = *(it-1) + 1 / cln::cl_N(i) * one;
        it++;
    }

    for (int n=1; n<ynsize; n++) {
        Yn[n].resize(newsize);
        std::vector<cln::cl_N>::iterator it = Yn[n].begin();
        std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
        it += ynlength;
        itprev += ynlength;
        for (int i=ynlength+n+1; i<=newsize+n; i++) {
            *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
            it++;
            itprev++;
        }
    }
    
    ynlength = newsize;
}


// helper function for S(n,p,x)
// [Kol] (7.2)
cln::cl_N C(int n, int p)
{
    cln::cl_N result;

    for (int k=0; k<p; k++) {
        for (int j=0; j<=(n+k-1)/2; j++) {
            if (k == 0) {
                if (n & 1) {
                    if (j & 1) {
                        result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
                    }
                    else {
                        result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
                    }
                }
            }
            else {
                if (k & 1) {
                    if (j & 1) {
                        result = result + cln::factorial(n+k-1)
                                          * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
                                          / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
                    }
                    else {
                        result = result - cln::factorial(n+k-1)
                                          * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
                                          / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
                    }
                }
                else {
                    if (j & 1) {
                        result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
                                          / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
                    }
                    else {
                        result = result + cln::factorial(n+k-1)
                                          * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
                                          / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
                    }
                }
            }
        }
    }
    int np = n+p;
    if ((np-1) & 1) {
        if (((np)/2+n) & 1) {
            result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
        }
        else {
            result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
        }
    }

    return result;
}


// helper function for S(n,p,x)
// [Kol] remark to (9.1)
cln::cl_N a_k(int k)
{
    cln::cl_N result;

    if (k == 0) {
        return 1;
    }

    result = result;
    for (int m=2; m<=k; m++) {
        result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
    }

    return -result / k;
}


// helper function for S(n,p,x)
// [Kol] remark to (9.1)
cln::cl_N b_k(int k)
{
    cln::cl_N result;

    if (k == 0) {
        return 1;
    }

    result = result;
    for (int m=2; m<=k; m++) {
        result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
    }

    return result / k;
}


// helper function for S(n,p,x)
cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
{
    static cln::float_format_t oldprec = cln::default_float_format;

    if (p==1) {
        return Li_projection(n+1, x, prec);
    }

    // precision has changed, we need to clear lookup table Yn
    if ( oldprec != prec ) {
        Yn.clear();
        ynsize = 0;
        ynlength = 100;
        oldprec = prec;
    }
        
    // check if precalculated values are sufficient
    if (p > ynsize+1) {
        for (int i=ynsize; i<p-1; i++) {
            fill_Yn(i, prec);
        }
    }

    // should be done otherwise
    cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
    cln::cl_N xf = x * one;
    //cln::cl_N xf = x * cln::cl_float(1, prec);

    cln::cl_N res;
    cln::cl_N resbuf;
    cln::cl_N factor = cln::expt(xf, p);
    int i = p;
    do {
        resbuf = res;
        if (i-p >= ynlength) {
            // make Yn longer
            make_Yn_longer(ynlength*2, prec);
        }
        res = res + factor / cln::expt(cln::cl_I(i),n+1) * Yn[p-2][i-p]; // should we check it? or rely on magic number? ...
        //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
        factor = factor * xf;
        i++;
    } while (res != resbuf);
    
    return res;
}


// helper function for S(n,p,x)
cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
{
    // [Kol] (5.3)
    if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {

        cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
                           * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);

        for (int s=0; s<n; s++) {
            cln::cl_N res2;
            for (int r=0; r<p; r++) {
                res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
                              * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
            }
            result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
        }

        return result;
    }
    
    return S_do_sum(n, p, x, prec);
}


// helper function for S(n,p,x)
const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
{
    if (x == 1) {
        if (n == 1) {
            // [Kol] (2.22) with (2.21)
            return cln::zeta(p+1);
        }

        if (p == 1) {
            // [Kol] (2.22)
            return cln::zeta(n+1);
        }

        // [Kol] (9.1)
        cln::cl_N result;
        for (int nu=0; nu<n; nu++) {
            for (int rho=0; rho<=p; rho++) {
                result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
                                  * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
            }
        }
        result = result * cln::expt(cln::cl_I(-1),n+p-1);

        return result;
    }
    else if (x == -1) {
        // [Kol] (2.22)
        if (p == 1) {
            return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
        }
//      throw std::runtime_error("don't know how to evaluate this function!");
    }

    // what is the desired float format?
    // first guess: default format
    cln::float_format_t prec = cln::default_float_format;
    const cln::cl_N value = x;
    // second guess: the argument's format
    if (!instanceof(realpart(value), cln::cl_RA_ring))
        prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
    else if (!instanceof(imagpart(value), cln::cl_RA_ring))
        prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));

    // [Kol] (5.3)
    if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95))) {

        cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
                           * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);

        for (int s=0; s<n; s++) {
            cln::cl_N res2;
            for (int r=0; r<p; r++) {
                res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
                              * S_num(p-r,n-s,1-value) / cln::factorial(r);
            }
            result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
        }

        return result;
        
    }
    // [Kol] (5.12)
    if (cln::abs(value) > 1) {
        
        cln::cl_N result;

        for (int s=0; s<p; s++) {
            for (int r=0; r<=s; r++) {
                result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
                                  / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
                                  * S_num(n+s-r,p-s,cln::recip(value));
            }
        }
        result = result * cln::expt(cln::cl_I(-1),n);

        cln::cl_N res2;
        for (int r=0; r<n; r++) {
            res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
        }
        res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);

        result = result + cln::expt(cln::cl_I(-1),p) * res2;

        return result;
    }
    else {
        return S_projection(n, p, value, prec);
    }
}


} // end of anonymous namespace


//
// Nielsen's generalized polylogarithm  S(n,p,x)
//
// GiNaC function
//


static ex S_evalf(const ex& n, const ex& p, const ex& x)
{
    if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
        const int n_ = ex_to<numeric>(n).to_int();
        const int p_ = ex_to<numeric>(p).to_int();
        if (is_a<numeric>(x)) {
            const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
            const cln::cl_N result = S_num(n_, p_, x_);
            return numeric(result);
        } else {
            ex x_val = x.evalf();
            if (is_a<numeric>(x_val)) {
                const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
                const cln::cl_N result = S_num(n_, p_, x_val_);
                return numeric(result);
            }
        }
    }
    return S(n, p, x).hold();
}


static ex S_eval(const ex& n, const ex& p, const ex& x)
{
    if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
        if (x == 0) {
            return _ex0;
        }
        if (x == 1) {
            lst m(n+1);
            for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
                m.append(1);
            }
            return zeta(m);
        }
        if (p == 1) {
            return Li(n+1, x);
        }
        if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
            int n_ = ex_to<numeric>(n).to_int();
            int p_ = ex_to<numeric>(p).to_int();
            const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
            const cln::cl_N result = S_num(n_, p_, x_);
            return numeric(result);
        }
    }
    if (n.is_zero()) {
        // [Kol] (5.3)
        return pow(-log(1-x), p) / factorial(p);
    }
    return S(n, p, x).hold();
}


static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
{
    if (p == _ex1) {
        return Li(n+1, x).series(rel, order, options);
    }

    const ex x_pt = x.subs(rel, subs_options::no_pattern);
    if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
        // First special case: x==0 (derivatives have poles)
        if (x_pt.is_zero()) {
            const symbol s;
            ex ser;
            // manually construct the primitive expansion
            // subsum = Euler-Zagier-Sum is needed
            // dirty hack (slow ...) calculation of subsum:
            std::vector<ex> presubsum, subsum;
            subsum.push_back(0);
            for (int i=1; i<order-1; ++i) {
                subsum.push_back(subsum[i-1] + numeric(1, i));
            }
            for (int depth=2; depth<p; ++depth) {
                presubsum = subsum;
                for (int i=1; i<order-1; ++i) {
                    subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
                }
            }
                
            for (int i=1; i<order; ++i) {
                ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
            }
            // substitute the argument's series expansion
            ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
            // maybe that was terminating, so add a proper order term
            epvector nseq;
            nseq.push_back(expair(Order(_ex1), order));
            ser += pseries(rel, nseq);
            // reexpanding it will collapse the series again
            return ser.series(rel, order);
        }
        // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
        throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
    }
    // all other cases should be safe, by now:
    throw do_taylor();  // caught by function::series()
}


static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param < 3);
    if (deriv_param < 2) {
        return _ex0;
    }
    if (n > 0) {
        return S(n-1, p, x) / x;
    } else {
        return S(n, p-1, x) / (1-x);
    }
}


static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
{
    c.s << "\\mathrm{S}_{";
    n.print(c);
    c.s << ",";
    p.print(c);
    c.s << "}(";
    x.print(c);
    c.s << ")";
}


REGISTER_FUNCTION(S,
                  evalf_func(S_evalf).
                  eval_func(S_eval).
                  series_func(S_series).
                  derivative_func(S_deriv).
                  print_func<print_latex>(S_print_latex).
                  do_not_evalf_params());


//
// Harmonic polylogarithm  H(m,x)
//
// helper functions
//


// anonymous namespace for helper functions
namespace {

    
// regulates the pole (used by 1/x-transformation)
symbol H_polesign("IMSIGN");


// convert parameters from H to Li representation
// parameters are expected to be in expanded form, i.e. only 0, 1 and -1
// returns true if some parameters are negative
bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
{
    // expand parameter list
    lst mexp;
    for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
        if (*it > 1) {
            for (ex count=*it-1; count > 0; count--) {
                mexp.append(0);
            }
            mexp.append(1);
        } else if (*it < -1) {
            for (ex count=*it+1; count < 0; count++) {
                mexp.append(0);
            }
            mexp.append(-1);
        } else {
            mexp.append(*it);
        }
    }
    
    ex signum = 1;
    pf = 1;
    bool has_negative_parameters = false;
    ex acc = 1;
    for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
        if (*it == 0) {
            acc++;
            continue;
        }
        if (*it > 0) {
            m.append((*it+acc-1) * signum);
        } else {
            m.append((*it-acc+1) * signum);
        }
        acc = 1;
        signum = *it;
        pf *= *it;
        if (pf < 0) {
            has_negative_parameters = true;
        }
    }
    if (has_negative_parameters) {
        for (std::size_t i=0; i<m.nops(); i++) {
            if (m.op(i) < 0) {
                m.let_op(i) = -m.op(i);
                s.append(-1);
            } else {
                s.append(1);
            }
        }
    }
    
    return has_negative_parameters;
}


// recursivly transforms H to corresponding multiple polylogarithms
struct map_trafo_H_convert_to_Li : public map_function
{
    ex operator()(const ex& e)
    {
        if (is_a<add>(e) || is_a<mul>(e)) {
            return e.map(*this);
        }
        if (is_a<function>(e)) {
            std::string name = ex_to<function>(e).get_name();
            if (name == "H") {
                lst parameter;
                if (is_a<lst>(e.op(0))) {
                        parameter = ex_to<lst>(e.op(0));
                } else {
                    parameter = lst(e.op(0));
                }
                ex arg = e.op(1);

                lst m;
                lst s;
                ex pf;
                if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
                    s.let_op(0) = s.op(0) * arg;
                    return pf * Li(m, s).hold();
                } else {
                    for (std::size_t i=0; i<m.nops(); i++) {
                        s.append(1);
                    }
                    s.let_op(0) = s.op(0) * arg;
                    return Li(m, s).hold();
                }
            }
        }
        return e;
    }
};


// recursivly transforms H to corresponding zetas
struct map_trafo_H_convert_to_zeta : public map_function
{
    ex operator()(const ex& e)
    {
        if (is_a<add>(e) || is_a<mul>(e)) {
            return e.map(*this);
        }
        if (is_a<function>(e)) {
            std::string name = ex_to<function>(e).get_name();
            if (name == "H") {
                lst parameter;
                if (is_a<lst>(e.op(0))) {
                        parameter = ex_to<lst>(e.op(0));
                } else {
                    parameter = lst(e.op(0));
                }

                lst m;
                lst s;
                ex pf;
                if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
                    return pf * zeta(m, s);
                } else {
                    return zeta(m);
                }
            }
        }
        return e;
    }
};


// remove trailing zeros from H-parameters
struct map_trafo_H_reduce_trailing_zeros : public map_function
{
    ex operator()(const ex& e)
    {
        if (is_a<add>(e) || is_a<mul>(e)) {
            return e.map(*this);
        }
        if (is_a<function>(e)) {
            std::string name = ex_to<function>(e).get_name();
            if (name == "H") {
                lst parameter;
                if (is_a<lst>(e.op(0))) {
                    parameter = ex_to<lst>(e.op(0));
                } else {
                    parameter = lst(e.op(0));
                }
                ex arg = e.op(1);
                if (parameter.op(parameter.nops()-1) == 0) {
                    
                    //
                    if (parameter.nops() == 1) {
                        return log(arg);
                    }
                    
                    //
                    lst::const_iterator it = parameter.begin();
                    while ((it != parameter.end()) && (*it == 0)) {
                        it++;
                    }
                    if (it == parameter.end()) {
                        return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
                    }
                    
                    //
                    parameter.remove_last();
                    std::size_t lastentry = parameter.nops();
                    while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
                        lastentry--;
                    }
                    
                    //
                    ex result = log(arg) * H(parameter,arg).hold();
                    ex acc = 0;
                    for (ex i=0; i<lastentry; i++) {
                        if (parameter[i] > 0) {
                            parameter[i]++;
                            result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
                            parameter[i]--;
                            acc = 0;
                        } else if (parameter[i] < 0) {
                            parameter[i]--;
                            result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
                            parameter[i]++;
                            acc = 0;
                        } else {
                            acc++;
                        }
                    }
                    
                    if (lastentry < parameter.nops()) {
                        result = result / (parameter.nops()-lastentry+1);
                        return result.map(*this);
                    } else {
                        return result;
                    }
                }
            }
        }
        return e;
    }
};


// returns an expression with zeta functions corresponding to the parameter list for H
ex convert_H_to_zeta(const lst& m)
{
    symbol xtemp("xtemp");
    map_trafo_H_reduce_trailing_zeros filter;
    map_trafo_H_convert_to_zeta filter2;
    return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
}


// convert signs form Li to H representation
lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
{
    lst res;
    lst::const_iterator itm = m.begin();
    lst::const_iterator itx = ++x.begin();
    int signum = 1;
    pf = _ex1;
    res.append(*itm);
    itm++;
    while (itx != x.end()) {
        signum *= (*itx > 0) ? 1 : -1;
        pf *= signum;
        res.append((*itm) * signum);
        itm++;
        itx++;
    }
    return res;
}


// multiplies an one-dimensional H with another H
// [ReV] (18)
ex trafo_H_mult(const ex& h1, const ex& h2)
{
    ex res;
    ex hshort;
    lst hlong;
    ex h1nops = h1.op(0).nops();
    ex h2nops = h2.op(0).nops();
    if (h1nops > 1) {
        hshort = h2.op(0).op(0);
        hlong = ex_to<lst>(h1.op(0));
    } else {
        hshort = h1.op(0).op(0);
        if (h2nops > 1) {
            hlong = ex_to<lst>(h2.op(0));
        } else {
            hlong = h2.op(0).op(0);
        }
    }
    for (std::size_t i=0; i<=hlong.nops(); i++) {
        lst newparameter;
        std::size_t j=0;
        for (; j<i; j++) {
            newparameter.append(hlong[j]);
        }
        newparameter.append(hshort);
        for (; j<hlong.nops(); j++) {
            newparameter.append(hlong[j]);
        }
        res += H(newparameter, h1.op(1)).hold();
    }
    return res;
}


// applies trafo_H_mult recursively on expressions
struct map_trafo_H_mult : public map_function
{
    ex operator()(const ex& e)
    {
        if (is_a<add>(e)) {
            return e.map(*this);
        }

        if (is_a<mul>(e)) {

            ex result = 1;
            ex firstH;
            lst Hlst;
            for (std::size_t pos=0; pos<e.nops(); pos++) {
                if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
                    std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
                    if (name == "H") {
                        for (ex i=0; i<e.op(pos).op(1); i++) {
                            Hlst.append(e.op(pos).op(0));
                        }
                        continue;
                    }
                } else if (is_a<function>(e.op(pos))) {
                    std::string name = ex_to<function>(e.op(pos)).get_name();
                    if (name == "H") {
                        if (e.op(pos).op(0).nops() > 1) {
                            firstH = e.op(pos);
                        } else {
                            Hlst.append(e.op(pos));
                        }
                        continue;
                    }
                }
                result *= e.op(pos);
            }
            if (firstH == 0) {
                if (Hlst.nops() > 0) {
                    firstH = Hlst[Hlst.nops()-1];
                    Hlst.remove_last();
                } else {
                    return e;
                }
            }

            if (Hlst.nops() > 0) {
                ex buffer = trafo_H_mult(firstH, Hlst.op(0));
                result *= buffer;
                for (std::size_t i=1; i<Hlst.nops(); i++) {
                    result *= Hlst.op(i);
                }
                result = result.expand();
                map_trafo_H_mult recursion;
                return recursion(result);
            } else {
                return e;
            }

        }
        return e;
    }
};


// do integration [ReV] (55)
// put parameter 0 in front of existing parameters
ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
{
    ex h;
    std::string name;
    if (is_a<function>(e)) {
        name = ex_to<function>(e).get_name();
    }
    if (name == "H") {
        h = e;
    } else {
        for (std::size_t i=0; i<e.nops(); i++) {
            if (is_a<function>(e.op(i))) {
                std::string name = ex_to<function>(e.op(i)).get_name();
                if (name == "H") {
                    h = e.op(i);
                }
            }
        }
    }
    if (h != 0) {
        lst newparameter = ex_to<lst>(h.op(0));
        newparameter.prepend(0);
        ex addzeta = convert_H_to_zeta(newparameter);
        return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
    } else {
        return e * (-H(lst(0),1/arg).hold());
    }
}


// do integration [ReV] (49)
// put parameter 1 in front of existing parameters
ex trafo_H_prepend_one(const ex& e, const ex& arg)
{
    ex h;
    std::string name;
    if (is_a<function>(e)) {
        name = ex_to<function>(e).get_name();
    }
    if (name == "H") {
        h = e;
    } else {
        for (std::size_t i=0; i<e.nops(); i++) {
            if (is_a<function>(e.op(i))) {
                std::string name = ex_to<function>(e.op(i)).get_name();
                if (name == "H") {
                    h = e.op(i);
                }
            }
        }
    }
    if (h != 0) {
        lst newparameter = ex_to<lst>(h.op(0));
        newparameter.prepend(1);
        return e.subs(h == H(newparameter, h.op(1)).hold());
    } else {
        return e * H(lst(1),1-arg).hold();
    }
}


// do integration [ReV] (55)
// put parameter -1 in front of existing parameters
ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
{
    ex h;
    std::string name;
    if (is_a<function>(e)) {
        name = ex_to<function>(e).get_name();
    }
    if (name == "H") {
        h = e;
    } else {
        for (std::size_t i=0; i<e.nops(); i++) {
            if (is_a<function>(e.op(i))) {
                std::string name = ex_to<function>(e.op(i)).get_name();
                if (name == "H") {
                    h = e.op(i);
                }
            }
        }
    }
    if (h != 0) {
        lst newparameter = ex_to<lst>(h.op(0));
        newparameter.prepend(-1);
        ex addzeta = convert_H_to_zeta(newparameter);
        return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
    } else {
        ex addzeta = convert_H_to_zeta(lst(-1));
        return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
    }
}


// do integration [ReV] (55)
// put parameter -1 in front of existing parameters
ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
{
    ex h;
    std::string name;
    if (is_a<function>(e)) {
        name = ex_to<function>(e).get_name();
    }
    if (name == "H") {
        h = e;
    } else {
        for (std::size_t i = 0; i < e.nops(); i++) {
            if (is_a<function>(e.op(i))) {
                std::string name = ex_to<function>(e.op(i)).get_name();
                if (name == "H") {
                    h = e.op(i);
                }
            }
        }
    }
    if (h != 0) {
        lst newparameter = ex_to<lst>(h.op(0));
        newparameter.prepend(-1);
        return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
    } else {
        return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
    }
}


// do integration [ReV] (55)
// put parameter 1 in front of existing parameters
ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
{
    ex h;
    std::string name;
    if (is_a<function>(e)) {
        name = ex_to<function>(e).get_name();
    }
    if (name == "H") {
        h = e;
    } else {
        for (std::size_t i = 0; i < e.nops(); i++) {
            if (is_a<function>(e.op(i))) {
                std::string name = ex_to<function>(e.op(i)).get_name();
                if (name == "H") {
                    h = e.op(i);
                }
            }
        }
    }
    if (h != 0) {
        lst newparameter = ex_to<lst>(h.op(0));
        newparameter.prepend(1);
        return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
    } else {
        return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
    }
}


// do x -> 1-x transformation
struct map_trafo_H_1mx : public map_function
{
    ex operator()(const ex& e)
    {
        if (is_a<add>(e) || is_a<mul>(e)) {
            return e.map(*this);
        }
        
        if (is_a<function>(e)) {
            std::string name = ex_to<function>(e).get_name();
            if (name == "H") {

                lst parameter = ex_to<lst>(e.op(0));
                ex arg = e.op(1);

                // special cases if all parameters are either 0, 1 or -1
                bool allthesame = true;
                if (parameter.op(0) == 0) {
                    for (std::size_t i = 1; i < parameter.nops(); i++) {
                        if (parameter.op(i) != 0) {
                            allthesame = false;
                            break;
                        }
                    }
                    if (allthesame) {
                        lst newparameter;
                        for (int i=parameter.nops(); i>0; i--) {
                            newparameter.append(1);
                        }
                        return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
                    }
                } else if (parameter.op(0) == -1) {
                    throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
                } else {
                    for (std::size_t i = 1; i < parameter.nops(); i++) {
                        if (parameter.op(i) != 1) {
                            allthesame = false;
                            break;
                        }
                    }
                    if (allthesame) {
                        lst newparameter;
                        for (int i=parameter.nops(); i>0; i--) {
                            newparameter.append(0);
                        }
                        return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
                    }
                }

                lst newparameter = parameter;
                newparameter.remove_first();

                if (parameter.op(0) == 0) {

                    // leading zero
                    ex res = convert_H_to_zeta(parameter);
                    //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
                    map_trafo_H_1mx recursion;
                    ex buffer = recursion(H(newparameter, arg).hold());
                    if (is_a<add>(buffer)) {
                        for (std::size_t i = 0; i < buffer.nops(); i++) {
                            res -= trafo_H_prepend_one(buffer.op(i), arg);
                        }
                    } else {
                        res -= trafo_H_prepend_one(buffer, arg);
                    }
                    return res;

                } else {

                    // leading one
                    map_trafo_H_1mx recursion;
                    map_trafo_H_mult unify;
                    ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
                    std::size_t firstzero = 0;
                    while (parameter.op(firstzero) == 1) {
                        firstzero++;
                    }
                    for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
                        lst newparameter;
                        std::size_t j=0;
                        for (; j<=i; j++) {
                            newparameter.append(parameter[j+1]);
                        }
                        newparameter.append(1);
                        for (; j<parameter.nops()-1; j++) {
                            newparameter.append(parameter[j+1]);
                        }
                        res -= H(newparameter, arg).hold();
                    }
                    res = recursion(res).expand() / firstzero;
                    return unify(res);
                }
            }
        }
        return e;
    }
};


// do x -> 1/x transformation
struct map_trafo_H_1overx : public map_function
{
    ex operator()(const ex& e)
    {
        if (is_a<add>(e) || is_a<mul>(e)) {
            return e.map(*this);
        }

        if (is_a<function>(e)) {
            std::string name = ex_to<function>(e).get_name();
            if (name == "H") {

                lst parameter = ex_to<lst>(e.op(0));
                ex arg = e.op(1);

                // special cases if all parameters are either 0, 1 or -1
                bool allthesame = true;
                if (parameter.op(0) == 0) {
                    for (std::size_t i = 1; i < parameter.nops(); i++) {
                        if (parameter.op(i) != 0) {
                            allthesame = false;
                            break;
                        }
                    }
                    if (allthesame) {
                        return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
                    }
                } else if (parameter.op(0) == -1) {
                    for (std::size_t i = 1; i < parameter.nops(); i++) {
                        if (parameter.op(i) != -1) {
                            allthesame = false;
                            break;
                        }
                    }
                    if (allthesame) {
                        map_trafo_H_mult unify;
                        return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
                               / factorial(parameter.nops())).expand());
                    }
                } else {
                    for (std::size_t i = 1; i < parameter.nops(); i++) {
                        if (parameter.op(i) != 1) {
                            allthesame = false;
                            break;
                        }
                    }
                    if (allthesame) {
                        map_trafo_H_mult unify;
                        return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
                               / factorial(parameter.nops())).expand());
                    }
                }

                lst newparameter = parameter;
                newparameter.remove_first();

                if (parameter.op(0) == 0) {
                    
                    // leading zero
                    ex res = convert_H_to_zeta(parameter);
                    map_trafo_H_1overx recursion;
                    ex buffer = recursion(H(newparameter, arg).hold());
                    if (is_a<add>(buffer)) {
                        for (std::size_t i = 0; i < buffer.nops(); i++) {
                            res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
                        }
                    } else {
                        res += trafo_H_1tx_prepend_zero(buffer, arg);
                    }
                    return res;

                } else if (parameter.op(0) == -1) {

                    // leading negative one
                    ex res = convert_H_to_zeta(parameter);
                    map_trafo_H_1overx recursion;
                    ex buffer = recursion(H(newparameter, arg).hold());
                    if (is_a<add>(buffer)) {
                        for (std::size_t i = 0; i < buffer.nops(); i++) {
                            res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
                        }
                    } else {
                        res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
                    }
                    return res;

                } else {

                    // leading one
                    map_trafo_H_1overx recursion;
                    map_trafo_H_mult unify;
                    ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
                    std::size_t firstzero = 0;
                    while (parameter.op(firstzero) == 1) {
                        firstzero++;
                    }
                    for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
                        lst newparameter;
                        std::size_t j = 0;
                        for (; j<=i; j++) {
                            newparameter.append(parameter[j+1]);
                        }
                        newparameter.append(1);
                        for (; j<parameter.nops()-1; j++) {
                            newparameter.append(parameter[j+1]);
                        }
                        res -= H(newparameter, arg).hold();
                    }
                    res = recursion(res).expand() / firstzero;
                    return unify(res);

                }

            }
        }
        return e;
    }
};


// do x -> (1-x)/(1+x) transformation
struct map_trafo_H_1mxt1px : public map_function
{
    ex operator()(const ex& e)
    {
        if (is_a<add>(e) || is_a<mul>(e)) {
            return e.map(*this);
        }

        if (is_a<function>(e)) {
            std::string name = ex_to<function>(e).get_name();
            if (name == "H") {

                lst parameter = ex_to<lst>(e.op(0));
                ex arg = e.op(1);

                // special cases if all parameters are either 0, 1 or -1
                bool allthesame = true;
                if (parameter.op(0) == 0) {
                    for (std::size_t i = 1; i < parameter.nops(); i++) {
                        if (parameter.op(i) != 0) {
                            allthesame = false;
                            break;
                        }
                    }
                    if (allthesame) {
                        map_trafo_H_mult unify;
                        return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
                               / factorial(parameter.nops())).expand());
                    }
                } else if (parameter.op(0) == -1) {
                    for (std::size_t i = 1; i < parameter.nops(); i++) {
                        if (parameter.op(i) != -1) {
                            allthesame = false;
                            break;
                        }
                    }
                    if (allthesame) {
                        map_trafo_H_mult unify;
                        return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
                               / factorial(parameter.nops())).expand());
                    }
                } else {
                    for (std::size_t i = 1; i < parameter.nops(); i++) {
                        if (parameter.op(i) != 1) {
                            allthesame = false;
                            break;
                        }
                    }
                    if (allthesame) {
                        map_trafo_H_mult unify;
                        return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
                               / factorial(parameter.nops())).expand());
                    }
                }

                lst newparameter = parameter;
                newparameter.remove_first();

                if (parameter.op(0) == 0) {

                    // leading zero
                    ex res = convert_H_to_zeta(parameter);
                    map_trafo_H_1mxt1px recursion;
                    ex buffer = recursion(H(newparameter, arg).hold());
                    if (is_a<add>(buffer)) {
                        for (std::size_t i = 0; i < buffer.nops(); i++) {
                            res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
                        }
                    } else {
                        res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
                    }
                    return res;

                } else if (parameter.op(0) == -1) {

                    // leading negative one
                    ex res = convert_H_to_zeta(parameter);
                    map_trafo_H_1mxt1px recursion;
                    ex buffer = recursion(H(newparameter, arg).hold());
                    if (is_a<add>(buffer)) {
                        for (std::size_t i = 0; i < buffer.nops(); i++) {
                            res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
                        }
                    } else {
                        res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
                    }
                    return res;

                } else {

                    // leading one
                    map_trafo_H_1mxt1px recursion;
                    map_trafo_H_mult unify;
                    ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
                    std::size_t firstzero = 0;
                    while (parameter.op(firstzero) == 1) {
                        firstzero++;
                    }
                    for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
                        lst newparameter;
                        std::size_t j=0;
                        for (; j<=i; j++) {
                            newparameter.append(parameter[j+1]);
                        }
                        newparameter.append(1);
                        for (; j<parameter.nops()-1; j++) {
                            newparameter.append(parameter[j+1]);
                        }
                        res -= H(newparameter, arg).hold();
                    }
                    res = recursion(res).expand() / firstzero;
                    return unify(res);

                }

            }
        }
        return e;
    }
};


// do the actual summation.
cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
{
    const int j = m.size();

    std::vector<cln::cl_N> t(j);

    cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
    cln::cl_N factor = cln::expt(x, j) * one;
    cln::cl_N t0buf;
    int q = 0;
    do {
        t0buf = t[0];
        q++;
        t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
        for (int k=j-2; k>=1; k--) {
            t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
        }
        t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
        factor = factor * x;
    } while (t[0] != t0buf);

    return t[0];
}


} // end of anonymous namespace


//
// Harmonic polylogarithm  H(m,x)
//
// GiNaC function
//


static ex H_evalf(const ex& x1, const ex& x2)
{
    if (is_a<lst>(x1)) {
        
        cln::cl_N x;
        if (is_a<numeric>(x2)) {
            x = ex_to<numeric>(x2).to_cl_N();
        } else {
            ex x2_val = x2.evalf();
            if (is_a<numeric>(x2_val)) {
                x = ex_to<numeric>(x2_val).to_cl_N();
            }
        }

        for (std::size_t i = 0; i < x1.nops(); i++) {
            if (!x1.op(i).info(info_flags::integer)) {
                return H(x1, x2).hold();
            }
        }
        if (x1.nops() < 1) {
            return H(x1, x2).hold();
        }

        const lst& morg = ex_to<lst>(x1);
        // remove trailing zeros ...
        if (*(--morg.end()) == 0) {
            symbol xtemp("xtemp");
            map_trafo_H_reduce_trailing_zeros filter;
            return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
        }
        // ... and expand parameter notation
        bool has_minus_one = false;
        lst m;
        for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
            if (*it > 1) {
                for (ex count=*it-1; count > 0; count--) {
                    m.append(0);
                }
                m.append(1);
            } else if (*it <= -1) {
                for (ex count=*it+1; count < 0; count++) {
                    m.append(0);
                }
                m.append(-1);
                has_minus_one = true;
            } else {
                m.append(*it);
            }
        }

        // do summation
        if (cln::abs(x) < 0.95) {
            lst m_lst;
            lst s_lst;
            ex pf;
            if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
                // negative parameters -> s_lst is filled
                std::vector<int> m_int;
                std::vector<cln::cl_N> x_cln;
                for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin(); 
                     it_int != m_lst.end(); it_int++, it_cln++) {
                    m_int.push_back(ex_to<numeric>(*it_int).to_int());
                    x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
                }
                x_cln.front() = x_cln.front() * x;
                return pf * numeric(multipleLi_do_sum(m_int, x_cln));
            } else {
                // only positive parameters
                //TODO
                if (m_lst.nops() == 1) {
                    return Li(m_lst.op(0), x2).evalf();
                }
                std::vector<int> m_int;
                for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
                    m_int.push_back(ex_to<numeric>(*it).to_int());
                }
                return numeric(H_do_sum(m_int, x));
            }
        }

        symbol xtemp("xtemp");
        ex res = 1; 
        
        // ensure that the realpart of the argument is positive
        if (cln::realpart(x) < 0) {
            x = -x;
            for (std::size_t i = 0; i < m.nops(); i++) {
                if (m.op(i) != 0) {
                    m.let_op(i) = -m.op(i);
                    res *= -1;
                }
            }
        }

        // x -> 1/x
        if (cln::abs(x) >= 2.0) {
            map_trafo_H_1overx trafo;
            res *= trafo(H(m, xtemp));
            if (cln::imagpart(x) <= 0) {
                res = res.subs(H_polesign == -I*Pi);
            } else {
                res = res.subs(H_polesign == I*Pi);
            }
            return res.subs(xtemp == numeric(x)).evalf();
        }
        
        // check transformations for 0.95 <= |x| < 2.0
        
        // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
        if (cln::abs(x-9.53) <= 9.47) {
            // x -> (1-x)/(1+x)
            map_trafo_H_1mxt1px trafo;
            res *= trafo(H(m, xtemp));
        } else {
            // x -> 1-x
            if (has_minus_one) {
                map_trafo_H_convert_to_Li filter;
                return filter(H(m, numeric(x)).hold()).evalf();
            }
            map_trafo_H_1mx trafo;
            res *= trafo(H(m, xtemp));
        }

        return res.subs(xtemp == numeric(x)).evalf();
    }

    return H(x1,x2).hold();
}


static ex H_eval(const ex& m_, const ex& x)
{
    lst m;
    if (is_a<lst>(m_)) {
        m = ex_to<lst>(m_);
    } else {
        m = lst(m_);
    }
    if (m.nops() == 0) {
        return _ex1;
    }
    ex pos1;
    ex pos2;
    ex n;
    ex p;
    int step = 0;
    if (*m.begin() > _ex1) {
        step++;
        pos1 = _ex0;
        pos2 = _ex1;
        n = *m.begin()-1;
        p = _ex1;
    } else if (*m.begin() < _ex_1) {
        step++;
        pos1 = _ex0;
        pos2 = _ex_1;
        n = -*m.begin()-1;
        p = _ex1;
    } else if (*m.begin() == _ex0) {
        pos1 = _ex0;
        n = _ex1;
    } else {
        pos1 = *m.begin();
        p = _ex1;
    }
    for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
        if ((*it).info(info_flags::integer)) {
            if (step == 0) {
                if (*it > _ex1) {
                    if (pos1 == _ex0) {
                        step = 1;
                        pos2 = _ex1;
                        n += *it-1;
                        p = _ex1;
                    } else {
                        step = 2;
                    }
                } else if (*it < _ex_1) {
                    if (pos1 == _ex0) {
                        step = 1;
                        pos2 = _ex_1;
                        n += -*it-1;
                        p = _ex1;
                    } else {
                        step = 2;
                    }
                } else {
                    if (*it != pos1) {
                        step = 1;
                        pos2 = *it;
                    }
                    if (*it == _ex0) {
                        n++;
                    } else {
                        p++;
                    }
                }
            } else if (step == 1) {
                if (*it != pos2) {
                    step = 2;
                } else {
                    if (*it == _ex0) {
                        n++;
                    } else {
                        p++;
                    }
                }
            }
        } else {
            // if some m_i is not an integer
            return H(m_, x).hold();
        }
    }
    if ((x == _ex1) && (*(--m.end()) != _ex0)) {
        return convert_H_to_zeta(m);
    }
    if (step == 0) {
        if (pos1 == _ex0) {
            // all zero
            if (x == _ex0) {
                return H(m_, x).hold();
            }
            return pow(log(x), m.nops()) / factorial(m.nops());
        } else {
            // all (minus) one
            return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
        }
    } else if ((step == 1) && (pos1 == _ex0)){
        // convertible to S
        if (pos2 == _ex1) {
            return S(n, p, x);
        } else {
            return pow(-1, p) * S(n, p, -x);
        }
    }
    if (x == _ex0) {
        return _ex0;
    }
    if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
        return H(m_, x).evalf();
    }
    return H(m_, x).hold();
}


static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
{
    epvector seq;
    seq.push_back(expair(H(m, x), 0));
    return pseries(rel, seq);
}


static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param < 2);
    if (deriv_param == 0) {
        return _ex0;
    }
    lst m;
    if (is_a<lst>(m_)) {
        m = ex_to<lst>(m_);
    } else {
        m = lst(m_);
    }
    ex mb = *m.begin();
    if (mb > _ex1) {
        m[0]--;
        return H(m, x) / x;
    }
    if (mb < _ex_1) {
        m[0]++;
        return H(m, x) / x;
    }
    m.remove_first();
    if (mb == _ex1) {
        return 1/(1-x) * H(m, x);
    } else if (mb == _ex_1) {
        return 1/(1+x) * H(m, x);
    } else {
        return H(m, x) / x;
    }
}


static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
{
    lst m;
    if (is_a<lst>(m_)) {
        m = ex_to<lst>(m_);
    } else {
        m = lst(m_);
    }
    c.s << "\\mathrm{H}_{";
    lst::const_iterator itm = m.begin();
    (*itm).print(c);
    itm++;
    for (; itm != m.end(); itm++) {
        c.s << ",";
        (*itm).print(c);
    }
    c.s << "}(";
    x.print(c);
    c.s << ")";
}


REGISTER_FUNCTION(H,
                  evalf_func(H_evalf).
                  eval_func(H_eval).
                  series_func(H_series).
                  derivative_func(H_deriv).
                  print_func<print_latex>(H_print_latex).
                  do_not_evalf_params());


// takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
ex convert_H_to_Li(const ex& m, const ex& x)
{
    map_trafo_H_reduce_trailing_zeros filter;
    map_trafo_H_convert_to_Li filter2;
    if (is_a<lst>(m)) {
        return filter2(filter(H(m, x).hold()));
    } else {
        return filter2(filter(H(lst(m), x).hold()));
    }
}


//
// Multiple zeta values  zeta(x) and zeta(x,s)
//
// helper functions
//


// anonymous namespace for helper functions
namespace {


// parameters and data for [Cra] algorithm
const cln::cl_N lambda = cln::cl_N("319/320");

void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
{
    const int size = a.size();
    for (int n=0; n<size; n++) {
        c[n] = 0;
        for (int m=0; m<=n; m++) {
            c[n] = c[n] + a[m]*b[n-m];
        }
    }
}


// [Cra] section 4
static void initcX(std::vector<cln::cl_N>& crX,
               const std::vector<int>& s,
           const int L2)
{
    std::vector<cln::cl_N> crB(L2 + 1);
    for (int i=0; i<=L2; i++)
        crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i);

    int Sm = 0;
    int Smp1 = 0;
    std::vector<std::vector<cln::cl_N> > crG(s.size() - 1, std::vector<cln::cl_N>(L2 + 1));
    for (int m=0; m < (int)s.size() - 1; m++) {
        Sm += s[m];
        Smp1 = Sm + s[m+1];
        for (int i = 0; i <= L2; i++)
            crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2);
    }

    crX = crB;

    for (std::size_t m = 0; m < s.size() - 1; m++) {
        std::vector<cln::cl_N> Xbuf(L2 + 1);
        for (int i = 0; i <= L2; i++)
            Xbuf[i] = crX[i] * crG[m][i];

        halfcyclic_convolute(Xbuf, crB, crX);
    }
}


// [Cra] section 4
static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk,
                             const std::vector<cln::cl_N>& crX)
{
    cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
    cln::cl_N factor = cln::expt(lambda, Sqk);
    cln::cl_N res = factor / Sqk * crX[0] * one;
    cln::cl_N resbuf;
    int N = 0;
    do {
        resbuf = res;
        factor = factor * lambda;
        N++;
        res = res + crX[N] * factor / (N+Sqk);
    } while ((res != resbuf) || cln::zerop(crX[N]));
    return res;
}


// [Cra] section 4
static void calc_f(std::vector<std::vector<cln::cl_N> >& f_kj,
               const int maxr, const int L1)
{
    cln::cl_N t0, t1, t2, t3, t4;
    int i, j, k;
    std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
    cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
    
    t0 = cln::exp(-lambda);
    t2 = 1;
    for (k=1; k<=L1; k++) {
        t1 = k * lambda;
        t2 = t0 * t2;
        for (j=1; j<=maxr; j++) {
            t3 = 1;
            t4 = 1;
            for (i=2; i<=j; i++) {
                t4 = t4 * (j-i+1);
                t3 = t1 * t3 + t4;
            }
            (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
        }
        it++;
    }
}


// [Cra] (3.1)
static cln::cl_N crandall_Z(const std::vector<int>& s,
                        const std::vector<std::vector<cln::cl_N> >& f_kj)
{
    const int j = s.size();

    if (j == 1) {   
        cln::cl_N t0;
        cln::cl_N t0buf;
        int q = 0;
        do {
            t0buf = t0;
            q++;
            t0 = t0 + f_kj[q+j-2][s[0]-1];
        } while (t0 != t0buf);
        
        return t0 / cln::factorial(s[0]-1);
    }

    std::vector<cln::cl_N> t(j);

    cln::cl_N t0buf;
    int q = 0;
    do {
        t0buf = t[0];
        q++;
        t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
        for (int k=j-2; k>=1; k--) {
            t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
        }
        t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
    } while (t[0] != t0buf);
    
    return t[0] / cln::factorial(s[0]-1);
}


// [Cra] (2.4)
cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
{
    std::vector<int> r = s;
    const int j = r.size();

    std::size_t L1;

    // decide on maximal size of f_kj for crandall_Z
    if (Digits < 50) {
        L1 = 150;
    } else {
        L1 = Digits * 3 + j*2;
    }

    std::size_t L2;
    // decide on maximal size of crX for crandall_Y
    if (Digits < 38) {
        L2 = 63;
    } else if (Digits < 86) {
        L2 = 127;
    } else if (Digits < 192) {
        L2 = 255;
    } else if (Digits < 394) {
        L2 = 511;
    } else if (Digits < 808) {
        L2 = 1023;
    } else {
        L2 = 2047;
    }

    cln::cl_N res;

    int maxr = 0;
    int S = 0;
    for (int i=0; i<j; i++) {
        S += r[i];
        if (r[i] > maxr) {
            maxr = r[i];
        }
    }

    std::vector<std::vector<cln::cl_N> > f_kj(L1);
    calc_f(f_kj, maxr, L1);

    const cln::cl_N r0factorial = cln::factorial(r[0]-1);

    std::vector<int> rz;
    int skp1buf;
    int Srun = S;
    for (int k=r.size()-1; k>0; k--) {

        rz.insert(rz.begin(), r.back());
        skp1buf = rz.front();
        Srun -= skp1buf;
        r.pop_back();

        std::vector<cln::cl_N> crX;
        initcX(crX, r, L2);
        
        for (int q=0; q<skp1buf; q++) {
            
            cln::cl_N pp1 = crandall_Y_loop(Srun+q-k, crX);
            cln::cl_N pp2 = crandall_Z(rz, f_kj);

            rz.front()--;
            
            if (q & 1) {
                res = res - pp1 * pp2 / cln::factorial(q);
            } else {
                res = res + pp1 * pp2 / cln::factorial(q);
            }
        }
        rz.front() = skp1buf;
    }
    rz.insert(rz.begin(), r.back());

    std::vector<cln::cl_N> crX;
    initcX(crX, rz, L2);

    res = (res + crandall_Y_loop(S-j, crX)) / r0factorial
        + crandall_Z(rz, f_kj);

    return res;
}


cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
{
    const int j = r.size();

    // buffer for subsums
    std::vector<cln::cl_N> t(j);
    cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));

    cln::cl_N t0buf;
    int q = 0;
    do {
        t0buf = t[0];
        q++;
        t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
        for (int k=j-2; k>=0; k--) {
            t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
        }
    } while (t[0] != t0buf);

    return t[0];
}


// does Hoelder convolution. see [BBB] (7.0)
cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
{
    // prepare parameters
    // holds Li arguments in [BBB] notation
    std::vector<int> s = s_;
    std::vector<int> m_p = m_;
    std::vector<int> m_q;
    // holds Li arguments in nested sums notation
    std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
    s_p[0] = s_p[0] * cln::cl_N("1/2");
    // convert notations
    int sig = 1;
    for (std::size_t i = 0; i < s_.size(); i++) {
        if (s_[i] < 0) {
            sig = -sig;
            s_p[i] = -s_p[i];
        }
        s[i] = sig * std::abs(s[i]);
    }
    std::vector<cln::cl_N> s_q;
    cln::cl_N signum = 1;

    // first term
    cln::cl_N res = multipleLi_do_sum(m_p, s_p);

    // middle terms
    do {

        // change parameters
        if (s.front() > 0) {
            if (m_p.front() == 1) {
                m_p.erase(m_p.begin());
                s_p.erase(s_p.begin());
                if (s_p.size() > 0) {
                    s_p.front() = s_p.front() * cln::cl_N("1/2");
                }
                s.erase(s.begin());
                m_q.front()++;
            } else {
                m_p.front()--;
                m_q.insert(m_q.begin(), 1);
                if (s_q.size() > 0) {
                    s_q.front() = s_q.front() * 2;
                }
                s_q.insert(s_q.begin(), cln::cl_N("1/2"));
            }
        } else {
            if (m_p.front() == 1) {
                m_p.erase(m_p.begin());
                cln::cl_N spbuf = s_p.front();
                s_p.erase(s_p.begin());
                if (s_p.size() > 0) {
                    s_p.front() = s_p.front() * spbuf;
                }
                s.erase(s.begin());
                m_q.insert(m_q.begin(), 1);
                if (s_q.size() > 0) {
                    s_q.front() = s_q.front() * 4;
                }
                s_q.insert(s_q.begin(), cln::cl_N("1/4"));
                signum = -signum;
            } else {
                m_p.front()--;
                m_q.insert(m_q.begin(), 1);
                if (s_q.size() > 0) {
                    s_q.front() = s_q.front() * 2;
                }
                s_q.insert(s_q.begin(), cln::cl_N("1/2"));
            }
        }

        // exiting the loop
        if (m_p.size() == 0) break;

        res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);

    } while (true);

    // last term
    res = res + signum * multipleLi_do_sum(m_q, s_q);

    return res;
}


} // end of anonymous namespace


//
// Multiple zeta values  zeta(x)
//
// GiNaC function
//


static ex zeta1_evalf(const ex& x)
{
    if (is_exactly_a<lst>(x) && (x.nops()>1)) {

        // multiple zeta value
        const int count = x.nops();
        const lst& xlst = ex_to<lst>(x);
        std::vector<int> r(count);

        // check parameters and convert them
        lst::const_iterator it1 = xlst.begin();
        std::vector<int>::iterator it2 = r.begin();
        do {
            if (!(*it1).info(info_flags::posint)) {
                return zeta(x).hold();
            }
            *it2 = ex_to<numeric>(*it1).to_int();
            it1++;
            it2++;
        } while (it2 != r.end());

        // check for divergence
        if (r[0] == 1) {
            return zeta(x).hold();
        }

        // decide on summation algorithm
        // this is still a bit clumsy
        int limit = (Digits>17) ? 10 : 6;
        if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
            return numeric(zeta_do_sum_Crandall(r));
        } else {
            return numeric(zeta_do_sum_simple(r));
        }
    }

    // single zeta value
    if (is_exactly_a<numeric>(x) && (x != 1)) {
        try {
            return zeta(ex_to<numeric>(x));
        } catch (const dunno &e) { }
    }

    return zeta(x).hold();
}


static ex zeta1_eval(const ex& m)
{
    if (is_exactly_a<lst>(m)) {
        if (m.nops() == 1) {
            return zeta(m.op(0));
        }
        return zeta(m).hold();
    }

    if (m.info(info_flags::numeric)) {
        const numeric& y = ex_to<numeric>(m);
        // trap integer arguments:
        if (y.is_integer()) {
            if (y.is_zero()) {
                return _ex_1_2;
            }
            if (y.is_equal(*_num1_p)) {
                return zeta(m).hold();
            }
            if (y.info(info_flags::posint)) {
                if (y.info(info_flags::odd)) {
                    return zeta(m).hold();
                } else {
                    return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
                }
            } else {
                if (y.info(info_flags::odd)) {
                    return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
                } else {
                    return _ex0;
                }
            }
        }
        // zeta(float)
        if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
            return zeta1_evalf(m);
        }
    }
    return zeta(m).hold();
}


static ex zeta1_deriv(const ex& m, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);

    if (is_exactly_a<lst>(m)) {
        return _ex0;
    } else {
        return zetaderiv(_ex1, m);
    }
}


static void zeta1_print_latex(const ex& m_, const print_context& c)
{
    c.s << "\\zeta(";
    if (is_a<lst>(m_)) {
        const lst& m = ex_to<lst>(m_);
        lst::const_iterator it = m.begin();
        (*it).print(c);
        it++;
        for (; it != m.end(); it++) {
            c.s << ",";
            (*it).print(c);
        }
    } else {
        m_.print(c);
    }
    c.s << ")";
}


unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
                                evalf_func(zeta1_evalf).
                                eval_func(zeta1_eval).
                                derivative_func(zeta1_deriv).
                                print_func<print_latex>(zeta1_print_latex).
                                do_not_evalf_params().
                                overloaded(2));


//
// Alternating Euler sum  zeta(x,s)
//
// GiNaC function
//


static ex zeta2_evalf(const ex& x, const ex& s)
{
    if (is_exactly_a<lst>(x)) {

        // alternating Euler sum
        const int count = x.nops();
        const lst& xlst = ex_to<lst>(x);
        const lst& slst = ex_to<lst>(s);
        std::vector<int> xi(count);
        std::vector<int> si(count);

        // check parameters and convert them
        lst::const_iterator it_xread = xlst.begin();
        lst::const_iterator it_sread = slst.begin();
        std::vector<int>::iterator it_xwrite = xi.begin();
        std::vector<int>::iterator it_swrite = si.begin();
        do {
            if (!(*it_xread).info(info_flags::posint)) {
                return zeta(x, s).hold();
            }
            *it_xwrite = ex_to<numeric>(*it_xread).to_int();
            if (*it_sread > 0) {
                *it_swrite = 1;
            } else {
                *it_swrite = -1;
            }
            it_xread++;
            it_sread++;
            it_xwrite++;
            it_swrite++;
        } while (it_xwrite != xi.end());

        // check for divergence
        if ((xi[0] == 1) && (si[0] == 1)) {
            return zeta(x, s).hold();
        }

        // use Hoelder convolution
        return numeric(zeta_do_Hoelder_convolution(xi, si));
    }

    return zeta(x, s).hold();
}


static ex zeta2_eval(const ex& m, const ex& s_)
{
    if (is_exactly_a<lst>(s_)) {
        const lst& s = ex_to<lst>(s_);
        for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
            if ((*it).info(info_flags::positive)) {
                continue;
            }
            return zeta(m, s_).hold();
        }
        return zeta(m);
    } else if (s_.info(info_flags::positive)) {
        return zeta(m);
    }

    return zeta(m, s_).hold();
}


static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
{
    GINAC_ASSERT(deriv_param==0);

    if (is_exactly_a<lst>(m)) {
        return _ex0;
    } else {
        if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
            return zetaderiv(_ex1, m);
        }
        return _ex0;
    }
}


static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
{
    lst m;
    if (is_a<lst>(m_)) {
        m = ex_to<lst>(m_);
    } else {
        m = lst(m_);
    }
    lst s;
    if (is_a<lst>(s_)) {
        s = ex_to<lst>(s_);
    } else {
        s = lst(s_);
    }
    c.s << "\\zeta(";
    lst::const_iterator itm = m.begin();
    lst::const_iterator its = s.begin();
    if (*its < 0) {
        c.s << "\\overline{";
        (*itm).print(c);
        c.s << "}";
    } else {
        (*itm).print(c);
    }
    its++;
    itm++;
    for (; itm != m.end(); itm++, its++) {
        c.s << ",";
        if (*its < 0) {
            c.s << "\\overline{";
            (*itm).print(c);
            c.s << "}";
        } else {
            (*itm).print(c);
        }
    }
    c.s << ")";
}


unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
                                evalf_func(zeta2_evalf).
                                eval_func(zeta2_eval).
                                derivative_func(zeta2_deriv).
                                print_func<print_latex>(zeta2_print_latex).
                                do_not_evalf_params().
                                overloaded(2));


} // namespace GiNaC