Fixed bug in convergence check with complex arguments.
[ginac.git] / ginac / inifcns_nstdsums.cpp
index 92d23c9fc4189ef78463d89cb23b95c5b1dec29e..4fe91ad391eeb7f41e40362c7fa67223339da3a6 100644 (file)
@@ -1,28 +1,47 @@
 /** @file inifcns_nstdsums.cpp
  *
  *  Implementation of some special functions that have a representation as nested sums.
- *  The functions are: 
+ *
+ *  The functions are:
  *    classical polylogarithm              Li(n,x)
- *    multiple polylogarithm               Li(lst(n_1,...,n_k),lst(x_1,...,x_k)
+ *    multiple polylogarithm               Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
  *    nielsen's generalized polylogarithm  S(n,p,x)
- *    harmonic polylogarithm               H(lst(m_1,...,m_k),x)
- *    multiple zeta value                  mZeta(lst(m_1,...,m_k))
+ *    harmonic polylogarithm               H(m,x) or H(lst(m_1,...,m_k),x)
+ *    multiple zeta value                  zeta(m) or zeta(lst(m_1,...,m_k))
+ *    alternating Euler sum                zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
  *
  *  Some remarks:
- *    - All formulae used can be looked up in the following publication:
- *      Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
- *      This document will be referenced as [Kol] throughout this source code.
- *    - Classical polylogarithms (Li) and nielsen's generalized polylogarithms (S) can be numerically
- *     evaluated in the whole complex plane. And of course, there is still room for speed optimizations ;-).
- *    - The calculation of classical polylogarithms is speed up by using Euler-MacLaurin summation (EuMac).
- *    - The remaining functions can only be numerically evaluated with arguments lying in the unit sphere
- *      at the moment. Sorry. The evaluation especially for mZeta is very slow ... better not use it
- *      right now.
- *    - The functions have no series expansion. To do it, you have to convert these functions
+ *
+ *    - All formulae used can be looked up in the following publications:
+ *      [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
+ *      [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
+ *      [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
+ *      [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
+ *
+ *    - The order of parameters and arguments of Li and zeta is defined according to the nested sums
+ *      representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
+ *      0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
+ *      number --- notation.
+ *
+ *    - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
+ *      the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product
+ *      x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers.
+ *      If you want to have an alternating Euler sum, you have to give the signs of the parameters as a
+ *      second argument s to zeta(m,s) containing 1 and -1.
+ *
+ *    - The calculation of classical polylogarithms is speed up by using Bernoulli numbers and 
+ *      look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
+ *      [Cra] and [BBB] for speed up.
+ *
+ *    - The functions have no series expansion into nested sums. To do this, you have to convert these functions
  *      into the appropriate objects from the nestedsums library, do the expansion and convert the
- *      result back. 
+ *      result back.
+ *
  *    - Numerical testing of this implementation has been performed by doing a comparison of results
- *      between this software and the commercial M.......... 4.1.
+ *      between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
+ *      by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
+ *      comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
+ *      around |x|=1 along with comparisons to corresponding zeta functions.
  *
  */
 
 #include <cln/cln.h>
 
 #include "inifcns.h"
+
+#include "add.h"
+#include "constant.h"
 #include "lst.h"
+#include "mul.h"
 #include "numeric.h"
 #include "operators.h"
-#include "relational.h"
+#include "power.h"
 #include "pseries.h"
+#include "relational.h"
+#include "symbol.h"
+#include "utils.h"
+#include "wildcard.h"
 
 
 namespace GiNaC {
 
-       
-// lookup table for Euler-MacLaurin optimization
-// see fill_Xn()
-std::vector<std::vector<cln::cl_N> > Xn;
-int xnsize = 0;
 
+//////////////////////////////////////////////////////////////////////
+//
+// Classical polylogarithm  Li(n,x)
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
 
-// lookup table for 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]
 
+// anonymous namespace for helper functions
+namespace {
 
-//////////////////////
-// helper functions //
-//////////////////////
+
+// lookup table for factors built from Bernoulli numbers
+// see fill_Xn()
+std::vector<std::vector<cln::cl_N> > Xn;
+int xnsize = 0;
 
 
-// This function calculates the X_n. The X_n are needed for the Euler-MacLaurin summation (EMS) of
-// classical polylogarithms.
-// With EMS the polylogs can be calculated as follows:
+// 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)
@@ -87,8 +114,8 @@ int ynlength = 100; // initial length of all Yn[i]
 // 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 EMS.
-static void fill_Xn(int n)
+// The second index in Xn corresponds to the index from the actual sum.
+void fill_Xn(int n)
 {
        // rule of thumb. needs to be improved. TODO
        const int initsize = Digits * 3 / 2;
@@ -157,88 +184,12 @@ static void fill_Xn(int n)
 }
 
 
-// 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}.
-static void fill_Yn(int n, const cln::float_format_t& prec)
-{
-       // TODO -> get rid of the magic number
-       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 ... 
-static 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;
-}
-
-
-// calculates Li(2,x) without EuMac
-static cln::cl_N Li2_series(const cln::cl_N& x)
+// 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_N num = x * cln::cl_float(1, cln::float_format(Digits));
        cln::cl_I den = 1; // n^2 = 1
        unsigned i = 3;
        do {
@@ -252,12 +203,12 @@ static cln::cl_N Li2_series(const cln::cl_N& x)
 }
 
 
-// calculates Li(2,x) with EuMac
-static cln::cl_N Li2_series_EuMac(const cln::cl_N& x)
+// 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();
        cln::cl_N u = -cln::log(1-x);
-       cln::cl_N factor = u;
+       cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
        cln::cl_N res = u - u*u/4;
        cln::cl_N resbuf;
        unsigned i = 1;
@@ -272,10 +223,10 @@ static cln::cl_N Li2_series_EuMac(const cln::cl_N& x)
 }
 
 
-// calculates Li(n,x), n>2 without EuMac
-static cln::cl_N Lin_series(int n, const cln::cl_N& x)
+// 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_N factor = x * cln::cl_float(1, cln::float_format(Digits));
        cln::cl_N res = x;
        cln::cl_N resbuf;
        int i=2;
@@ -289,12 +240,12 @@ static cln::cl_N Lin_series(int n, const cln::cl_N& x)
 }
 
 
-// calculates Li(n,x), n>2 with EuMac
-static cln::cl_N Lin_series_EuMac(int n, const cln::cl_N& x)
+// 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();
        cln::cl_N u = -cln::log(1-x);
-       cln::cl_N factor = u;
+       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;
@@ -310,11 +261,11 @@ static cln::cl_N Lin_series_EuMac(int n, const cln::cl_N& x)
 
 
 // forward declaration needed by function Li_projection and C below
-static numeric S_num(int n, int p, const numeric& x);
+numeric S_num(int n, int p, const numeric& x);
 
 
 // helper function for classical polylog Li
-static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
+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) {
@@ -330,16 +281,16 @@ static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_forma
                        // 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_series(x);
+                               return Li2_do_sum(x);
                        } else {
-                               return Li2_series_EuMac(x);
+                               return Li2_do_sum_Xn(x);
                        }
                } else {
                        // choose the faster algorithm
                        if (cln::abs(cln::realpart(x)) > 0.75) {
-                               return -Li2_series(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
+                               return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
                        } else {
-                               return -Li2_series_EuMac(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
+                               return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
                        }
                }
        } else {
@@ -352,17 +303,17 @@ static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_forma
 
                if (cln::realpart(x) < 0.5) {
                        // choose the faster algorithm
-                       // with n>=12 the "normal" summation always wins against EuMac
+                       // 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_series(n, x);
+                               return Lin_do_sum(n, x);
                        } else {
-                               return Lin_series_EuMac(n, x);
+                               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).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
-                                       * cln::expt(cln::log(x), j) / cln::factorial(j);
+                                                 * cln::expt(cln::log(x), j) / cln::factorial(j);
                        }
                        return result;
                }
@@ -371,7 +322,7 @@ static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_forma
 
 
 // helper function for classical polylog Li
-static numeric Li_num(int n, const numeric& x)
+numeric Li_num(int n, const numeric& x)
 {
        if (n == 1) {
                // just a log
@@ -422,7 +373,7 @@ static numeric Li_num(int n, const numeric& x)
                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))
-                                       * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
+                                   * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
                }
                result = result - add;
                return result;
@@ -433,187 +384,522 @@ static numeric Li_num(int n, const numeric& x)
 }
 
 
-// helper function for S(n,p,x)
-static cln::cl_N numeric_nielsen(int n, int step)
-{
-       if (step) {
-               cln::cl_N res;
-               for (int i=1; i<n; i++) {
-                       res = res + numeric_nielsen(i, step-1) / cln::cl_I(i);
-               }
-               return res;
-       }
-       else {
-               return 1;
-       }
-}
+} // end of anonymous namespace
 
 
-// helper function for S(n,p,x)
-// [Kol] (7.2)
-static cln::cl_N C(int n, int p)
-{
-       cln::cl_N result;
+//////////////////////////////////////////////////////////////////////
+//
+// Multiple polylogarithm  Li(n,x)
+//
+// helper function
+//
+//////////////////////////////////////////////////////////////////////
 
-       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).to_cl_N() / cln::factorial(2*j);
-                                       }
-                                       else {
-                                               result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / 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).to_cl_N()
-                                                       / (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).to_cl_N()
-                                                       / (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).to_cl_N()
-                                                       / (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).to_cl_N()
-                                                       / (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;
-}
+// anonymous namespace for helper function
+namespace {
 
 
-// helper function for S(n,p,x)
-// [Kol] remark to (9.1)
-static cln::cl_N a_k(int k)
+cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
 {
-       cln::cl_N result;
+       const int j = s.size();
 
-       if (k == 0) {
-               return 1;
-       }
+       std::vector<cln::cl_N> t(j);
+       cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
 
-       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);
-       }
+       cln::cl_N t0buf;
+       int q = 0;
+       do {
+               t0buf = t[0];
+               // do it once ...
+               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]);
+               }
+               // ... and do it again (to avoid premature drop out due to special arguments)
+               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]);
+               }
+       } while (t[0] != t0buf);
 
-       return -result / k;
+       return t[0];
 }
 
+// forward declaration for Li_eval()
+lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
 
-// helper function for S(n,p,x)
-// [Kol] remark to (9.1)
-static cln::cl_N b_k(int k)
-{
-       cln::cl_N result;
 
-       if (k == 0) {
-               return 1;
-       }
+} // end of anonymous namespace
 
-       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;
-}
+//////////////////////////////////////////////////////////////////////
+//
+// Classical polylogarithm and multiple polylogarithm  Li(n,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
 
 
-// helper function for S(n,p,x)
-static cln::cl_N S_series(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+static ex Li_evalf(const ex& x1, const ex& x2)
 {
-       if (p==1) {
-               return Li_projection(n+1, x, prec);
+       // classical polylogs
+       if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
+               return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
        }
-       
-       // TODO -> check for vector boundaries and do missing calculations
+       // multiple polylogs
+       else if (is_a<lst>(x1) && is_a<lst>(x2)) {
+               ex conv = 1;
+               for (int i=0; i<x1.nops(); i++) {
+                       if (!x1.op(i).info(info_flags::posint)) {
+                               return Li(x1, x2).hold();
+                       }
+                       if (!is_a<numeric>(x2.op(i))) {
+                               return Li(x1, x2).hold();
+                       }
+                       conv *= x2.op(i);
+                       if (abs(conv) >= 1) {
+                               return Li(x1, x2).hold();
+                       }
+               }
 
-       // check if precalculated values are sufficient
-       if (p > ynsize+1) {
-               for (int i=ynsize; i<p-1; i++) {
-                       fill_Yn(i, prec);
+               std::vector<int> m;
+               std::vector<cln::cl_N> x;
+               for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
+                       m.push_back(ex_to<numeric>(x1.op(i)).to_int());
+                       x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
                }
+
+               return numeric(multipleLi_do_sum(m, x));
        }
 
-       // should be done otherwise
-       cln::cl_N xf = x * cln::cl_float(1, prec);
+       return Li(x1,x2).hold();
+}
 
-       cln::cl_N result;
-       cln::cl_N resultbuffer;
-       int i;
-       for (i=p; true; i++) {
-               resultbuffer = result;
-               if (i-p >= ynlength) {
-                       // make Yn longer
-                       make_Yn_longer(ynlength*2, prec);
+
+static ex Li_eval(const ex& m_, const ex& x_)
+{
+       if (m_.nops() < 2) {
+               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 (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.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
+                       return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
+               }
+       } else {
+               bool ish = true;
+               bool iszeta = true;
+               bool iszero = false;
+               bool doevalf = false;
+               bool doevalfveto = true;
+               const lst& m = ex_to<lst>(m_);
+               const lst& x = ex_to<lst>(x_);
+               lst::const_iterator itm = m.begin();
+               lst::const_iterator itx = x.begin();
+               for (; 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()) {
+                                       ish = false;
+                               }
+                               iszeta = false;
+                       }
+                       if (*itx == _ex0) {
+                               iszero = true;
+                       }
+                       if (!(*itx).info(info_flags::numeric)) {
+                               doevalfveto = false;
+                       }
+                       if (!(*itx).info(info_flags::crational)) {
+                               doevalf = true;
+                       }
                }
-               result = result + cln::expt(xf,i) / cln::expt(cln::cl_I(i),n+1) * Yn[p-2][i-p]; // should we check it? or rely on magic number? ...
-               if (cln::zerop(result-resultbuffer)) {
-                       break;
+               if (iszeta) {
+                       return zeta(m_, x_);
+               }
+               if (iszero) {
+                       return _ex0;
+               }
+               if (ish) {
+                       ex pf;
+                       lst newm = convert_parameter_Li_to_H(m, x, pf);
+                       return pf * H(newm, x[0]);
+               }
+               if (doevalfveto && doevalf) {
+                       return Li(m_, x_).evalf();
                }
        }
-       
-       return result;
+       return Li(m_, x_).hold();
 }
 
 
-// helper function for S(n,p,x)
-static cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
 {
-       // [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);
+       epvector seq;
+       seq.push_back(expair(Li(m, x), 0));
+       return pseries(rel, seq);
+}
 
-               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_series(p-r,n-s,1-x,prec) / cln::factorial(r);
-                       }
-                       result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
-               }
 
-               return result;
+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);
        }
-       
-       return S_series(n, p, x, prec);
 }
 
 
-// helper function for S(n,p,x)
-static numeric S_num(int n, int p, const numeric& x)
+static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
 {
-       if (x == 1) {
-               if (n == 1) {
+       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 << "\\mbox{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).to_cl_N() / cln::factorial(2*j);
+                                       }
+                                       else {
+                                               result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / 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).to_cl_N()
+                                                                 / (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).to_cl_N()
+                                                                 / (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).to_cl_N()
+                                                                 / (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).to_cl_N()
+                                                                 / (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)
+{
+       if (p==1) {
+               return Li_projection(n+1, x, 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).to_cl_N() - res2) / cln::factorial(s);
+               }
+
+               return result;
+       }
+       
+       return S_do_sum(n, p, x, prec);
+}
+
+
+// helper function for S(n,p,x)
+numeric S_num(int n, int p, const numeric& x)
+{
+       if (x == 1) {
+               if (n == 1) {
                    // [Kol] (2.22) with (2.21)
                        return cln::zeta(p+1);
                }
@@ -628,7 +914,7 @@ static numeric S_num(int n, int p, const numeric& x)
                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);
+                                                 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
                        }
                }
                result = result * cln::expt(cln::cl_I(-1),n+p-1);
@@ -658,311 +944,1829 @@ static numeric S_num(int n, int p, const numeric& x)
        if (cln::realpart(value) < -0.5) {
 
                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);
+                                  * 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).to_cl_N() / cln::factorial(r);
+                                             * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
+                       }
+                       result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - 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)).to_cl_N();
+                       }
+               }
+               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) && is_a<numeric>(x)) {
+               return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
+       }
+       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))) {
+                       return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
+               }
+       }
+       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)
+{
+       epvector seq;
+       seq.push_back(expair(S(n, p, x), 0));
+       return pseries(rel, seq);
+}
+
+
+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 << "\\mbox{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 {
+
+
+// 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 (int 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 (int 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();
+                                       int 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();
+       ex signum = _ex1;
+       pf = _ex1;
+       res.append(*itm);
+       itm++;
+       while (itx != x.end()) {
+               signum *= *itx;
+               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 (int i=0; i<=hlong.nops(); i++) {
+               lst newparameter;
+               int 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 (int 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 (int 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 (int 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] (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 (int 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 (int 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 (int 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_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 (int 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 (int 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 (int 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() - I*Pi, 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 (int 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 (int 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();
+                                       int firstzero = 0;
+                                       while (parameter.op(firstzero) == 1) {
+                                               firstzero++;
+                                       }
+                                       for (int i=firstzero-1; i<parameter.nops()-1; i++) {
+                                               lst newparameter;
+                                               int 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 (int 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 (int 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 (int 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 (int 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 (int 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();
+                                       int firstzero = 0;
+                                       while (parameter.op(firstzero) == 1) {
+                                               firstzero++;
+                                       }
+                                       for (int i=firstzero-1; i<parameter.nops()-1; i++) {
+                                               lst newparameter;
+                                               int 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) && is_a<numeric>(x2)) {
+               for (int 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();
+               }
+
+               cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
+               
+               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
+               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);
+                       } else {
+                               m.append(*it);
+                       }
+               }
+
+               // since the transformations produce a lot of terms, they are only efficient for
+               // argument near one.
+               // no transformation needed -> 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));
                        }
-                       result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
                }
 
-               return result;
-               
-       }
-       // [Kol] (5.12)
-       if (cln::abs(value) > 1) {
+               ex res = 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)).to_cl_N();
+               // ensure that the realpart of the argument is positive
+               if (cln::realpart(x) < 0) {
+                       x = -x;
+                       for (int i=0; i<m.nops(); i++) {
+                               if (m.op(i) != 0) {
+                                       m.let_op(i) = -m.op(i);
+                                       res *= -1;
+                               }
                        }
                }
-               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);
+               // choose transformations
+               symbol xtemp("xtemp");
+               if (cln::abs(x-1) < 1.4142) {
+                       // x -> (1-x)/(1+x)
+                       map_trafo_H_1mxt1px trafo;
+                       res *= trafo(H(m, xtemp));
+               } else {
+                       // x -> 1/x
+                       map_trafo_H_1overx trafo;
+                       res *= trafo(H(m, xtemp));
                }
-               res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
 
-               result = result + cln::expt(cln::cl_I(-1),p) * res2;
+               // simplify result
+// TODO
+//             map_trafo_H_convert converter;
+//             res = converter(res).expand();
+//             lst ll;
+//             res.find(H(wild(1),wild(2)), ll);
+//             res.find(zeta(wild(1)), ll);
+//             res.find(zeta(wild(1),wild(2)), ll);
+//             res = res.collect(ll);
+
+               return res.subs(xtemp == numeric(x)).evalf();
+       }
 
-               return result;
+       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_);
        }
-       else {
-               return S_projection(n, p, value, prec);
+       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();
 }
 
 
-// helper function for multiple polylogarithm
-static cln::cl_N numeric_zsum(int n, std::vector<cln::cl_N>& x, std::vector<cln::cl_N>& m)
+static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
 {
-       cln::cl_N res;
-       if (x.empty()) {
-               return 1;
+       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_);
        }
-       for (int i=1; i<n; i++) {
-               std::vector<cln::cl_N>::iterator be;
-               std::vector<cln::cl_N>::iterator en;
-               be = x.begin();
-               be++;
-               en = x.end();
-               std::vector<cln::cl_N> xbuf(be, en);
-               be = m.begin();
-               be++;
-               en = m.end();
-               std::vector<cln::cl_N> mbuf(be, en);
-               res = res + cln::expt(x[0],i) / cln::expt(i,m[0]) * numeric_zsum(i, xbuf, mbuf);
+       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;
        }
-       return res;
 }
 
 
-// helper function for harmonic polylogarithm
-static cln::cl_N numeric_harmonic(int n, std::vector<cln::cl_N>& m)
+static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
 {
-       cln::cl_N res;
-       if (m.empty()) {
-               return 1;
+       lst m;
+       if (is_a<lst>(m_)) {
+               m = ex_to<lst>(m_);
+       } else {
+               m = lst(m_);
        }
-       for (int i=1; i<n; i++) {
-               std::vector<cln::cl_N>::iterator be;
-               std::vector<cln::cl_N>::iterator en;
-               be = m.begin();
-               be++;
-               en = m.end();
-               std::vector<cln::cl_N> mbuf(be, en);
-               res = res + cln::recip(cln::expt(i,m[0])) * numeric_harmonic(i, mbuf);
+       c.s << "\\mbox{H}_{";
+       lst::const_iterator itm = m.begin();
+       (*itm).print(c);
+       itm++;
+       for (; itm != m.end(); itm++) {
+               c.s << ",";
+               (*itm).print(c);
        }
-       return res;
+       c.s << "}(";
+       x.print(c);
+       c.s << ")";
 }
 
 
-/////////////////////////////
-// end of helper functions //
-/////////////////////////////
-
+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());
 
-// Polylogarithm and multiple polylogarithm
 
-static ex Li_eval(const ex& x1, const ex& x2)
+// 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)
 {
-       if (x2.is_zero()) {
-               return 0;
+       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()));
        }
-       else {
-               if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational)))
-                       return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
-               return Li(x1,x2).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");
+int L1;
+int L2;
+std::vector<std::vector<cln::cl_N> > f_kj;
+std::vector<cln::cl_N> crB;
+std::vector<std::vector<cln::cl_N> > crG;
+std::vector<cln::cl_N> crX;
+
+
+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];
+               }
        }
 }
 
-static ex Li_evalf(const ex& x1, const ex& x2)
+
+// [Cra] section 4
+void initcX(const std::vector<int>& s)
 {
-       // classical polylogs
-       if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
-               return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
+       const int k = s.size();
+
+       crX.clear();
+       crG.clear();
+       crB.clear();
+
+       for (int i=0; i<=L2; i++) {
+               crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
        }
-       // multiple polylogs
-       else if (is_a<lst>(x1) && is_a<lst>(x2)) {
-               for (int i=0; i<x1.nops(); i++) {
-                       if (!is_a<numeric>(x1.op(i)))
-                               return Li(x1,x2).hold();
-                       if (!is_a<numeric>(x2.op(i)))
-                               return Li(x1,x2).hold();
-                       if (x2.op(i) >= 1)
-                               return Li(x1,x2).hold();
+
+       int Sm = 0;
+       int Smp1 = 0;
+       for (int m=0; m<k-1; m++) {
+               std::vector<cln::cl_N> crGbuf;
+               Sm = Sm + s[m];
+               Smp1 = Sm + s[m+1];
+               for (int i=0; i<=L2; i++) {
+                       crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
                }
+               crG.push_back(crGbuf);
+       }
 
-               cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).to_cl_N();
-               cln::cl_N x_1 = ex_to<numeric>(x2.op(x2.nops()-1)).to_cl_N();
-               std::vector<cln::cl_N> x;
-               std::vector<cln::cl_N> m;
-               const int nops = ex_to<numeric>(x1.nops()).to_int();
-               for (int i=nops-2; i>=0; i--) {
-                       m.push_back(ex_to<numeric>(x1.op(i)).to_cl_N());
-                       x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
+       crX = crB;
+
+       for (int m=0; m<k-1; m++) {
+               std::vector<cln::cl_N> Xbuf;
+               for (int i=0; i<=L2; i++) {
+                       Xbuf.push_back(crX[i] * crG[m][i]);
                }
+               halfcyclic_convolute(Xbuf, crB, crX);
+       }
+}
+
+
+// [Cra] section 4
+cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
+{
+       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;
+}
+
 
-               cln::cl_N res;
-               cln::cl_N resbuf;
-               for (int i=nops; true; i++) {
-                       resbuf = res;
-                       res = res + cln::expt(x_1,i) / cln::expt(i,m_1) * numeric_zsum(i, x, m);
-                       if (cln::zerop(res-resbuf))
-                               break;
+// [Cra] section 4
+void calc_f(int maxr)
+{
+       f_kj.clear();
+       f_kj.resize(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++;
+       }
+}
 
-               return numeric(res);
 
+// [Cra] (3.1)
+cln::cl_N crandall_Z(const std::vector<int>& s)
+{
+       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);
        }
 
-       return Li(x1,x2).hold();
+       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);
 }
 
-static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
+
+// [Cra] (2.4)
+cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
 {
-       epvector seq;
-       seq.push_back(expair(Li(x1,x2), 0));
-       return pseries(rel,seq);
-}
+       std::vector<int> r = s;
+       const int j = r.size();
 
-REGISTER_FUNCTION(Li, eval_func(Li_eval).evalf_func(Li_evalf).do_not_evalf_params().series_func(Li_series));
+       // decide on maximal size of f_kj for crandall_Z
+       if (Digits < 50) {
+               L1 = 150;
+       } else {
+               L1 = Digits * 3 + j*2;
+       }
 
+       // 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;
+       }
 
-// Nielsen's generalized polylogarithm
+       cln::cl_N res;
 
-static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
-{
-       if (x2 == 1) {
-               return Li(x1+1,x3);
+       int maxr = 0;
+       int S = 0;
+       for (int i=0; i<j; i++) {
+               S += r[i];
+               if (r[i] > maxr) {
+                       maxr = r[i];
+               }
        }
-       if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) && 
-                       x1.info(info_flags::posint) && x2.info(info_flags::posint)) {
-               return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
+
+       calc_f(maxr);
+
+       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();
+
+               initcX(r);
+               
+               for (int q=0; q<skp1buf; q++) {
+                       
+                       cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
+                       cln::cl_N pp2 = crandall_Z(rz);
+
+                       rz.front()--;
+                       
+                       if (q & 1) {
+                               res = res - pp1 * pp2 / cln::factorial(q);
+                       } else {
+                               res = res + pp1 * pp2 / cln::factorial(q);
+                       }
+               }
+               rz.front() = skp1buf;
        }
-       return S(x1,x2,x3).hold();
+       rz.insert(rz.begin(), r.back());
+
+       initcX(rz);
+
+       res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
+
+       return res;
 }
 
-static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
+
+cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
 {
-       if (is_a<numeric>(x1) && is_a<numeric>(x2) && is_a<numeric>(x3)) {
-               if ((x3 == -1) && (x2 != 1)) {
-                       // no formula to evaluate this ... sorry
-//                     return S(x1,x2,x3).hold();
+       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]);
                }
-               return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
-       }
-       return S(x1,x2,x3).hold();
+       } while (t[0] != t0buf);
+
+       return t[0];
 }
 
-static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options)
+
+// does Hoelder convolution. see [BBB] (7.0)
+cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
 {
-       epvector seq;
-       seq.push_back(expair(S(x1,x2,x3), 0));
-       return pseries(rel,seq);
+       // 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 (int 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;
 }
 
-REGISTER_FUNCTION(S, eval_func(S_eval).evalf_func(S_evalf).do_not_evalf_params().series_func(S_series));
+
+} // end of anonymous namespace
 
 
-// Harmonic polylogarithm
+//////////////////////////////////////////////////////////////////////
+//
+// Multiple zeta values  zeta(x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
 
-static ex H_eval(const ex& x1, const ex& x2)
+
+static ex zeta1_evalf(const ex& x)
 {
-       if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) {
-               return H(x1,x2).evalf();
+       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));
+               }
        }
-       return H(x1,x2).hold();
+
+       // 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 H_evalf(const ex& x1, const ex& x2)
+
+static ex zeta1_eval(const ex& m)
 {
-       if (is_a<lst>(x1) && is_a<numeric>(x2)) {
-               for (int i=0; i<x1.nops(); i++) {
-                       if (!is_a<numeric>(x1.op(i)))
-                               return H(x1,x2).hold();
-               }
-               if (x2 >= 1) {
-                       return H(x1,x2).hold();
+       if (is_exactly_a<lst>(m)) {
+               if (m.nops() == 1) {
+                       return zeta(m.op(0));
                }
+               return zeta(m).hold();
+       }
 
-               cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).to_cl_N();
-               cln::cl_N x_1 = ex_to<numeric>(x2).to_cl_N();
-               std::vector<cln::cl_N> m;
-               const int nops = ex_to<numeric>(x1.nops()).to_int();
-               for (int i=nops-2; i>=0; i--) {
-                       m.push_back(ex_to<numeric>(x1.op(i)).to_cl_N());
+       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)) {
+                               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, y-_num1) / factorial(y);
+                               }
+                       } else {
+                               if (y.info(info_flags::odd)) {
+                                       return -bernoulli(_num1-y) / (_num1-y);
+                               } else {
+                                       return _ex0;
+                               }
+                       }
                }
-
-               cln::cl_N res;
-               cln::cl_N resbuf;
-               for (int i=nops; true; i++) {
-                       resbuf = res;
-                       res = res + cln::expt(x_1,i) / cln::expt(i,m_1) * numeric_harmonic(i, m);
-                       if (cln::zerop(res-resbuf))
-                               break;
+               // zeta(float)
+               if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
+                       return zeta1_evalf(m);
                }
+       }
+       return zeta(m).hold();
+}
 
-               return numeric(res);
 
-       }
+static ex zeta1_deriv(const ex& m, unsigned deriv_param)
+{
+       GINAC_ASSERT(deriv_param==0);
 
-       return H(x1,x2).hold();
+       if (is_exactly_a<lst>(m)) {
+               return _ex0;
+       } else {
+               return zetaderiv(_ex1, m);
+       }
 }
 
-static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
+
+static void zeta1_print_latex(const ex& m_, const print_context& c)
 {
-       epvector seq;
-       seq.push_back(expair(H(x1,x2), 0));
-       return pseries(rel,seq);
+       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 << ")";
 }
 
-REGISTER_FUNCTION(H, eval_func(H_eval).evalf_func(H_evalf).do_not_evalf_params().series_func(H_series));
 
+unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta").
+                                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));
 
-// Multiple zeta value
 
-static ex mZeta_eval(const ex& x1)
-{
-       return mZeta(x1).hold();
-}
+//////////////////////////////////////////////////////////////////////
+//
+// Alternating Euler sum  zeta(x,s)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
 
-static ex mZeta_evalf(const ex& x1)
-{
-       if (is_a<lst>(x1)) {
-               for (int i=0; i<x1.nops(); i++) {
-                       if (!is_a<numeric>(x1.op(i)))
-                               return mZeta(x1).hold();
-               }
 
-               cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).to_cl_N();
+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 (m_1 == 1) {
-                       return mZeta(x1).hold();
-               }
-               
-               std::vector<cln::cl_N> m;
-               const int nops = ex_to<numeric>(x1.nops()).to_int();
-               for (int i=nops-2; i>=0; i--) {
-                       m.push_back(ex_to<numeric>(x1.op(i)).to_cl_N());
+               if ((xi[0] == 1) && (si[0] == 1)) {
+                       return zeta(x, s).hold();
                }
 
-               cln::float_format_t prec = cln::default_float_format;
-               cln::cl_N res = cln::complex(cln::cl_float(0, prec), 0);
-               cln::cl_N resbuf;
-               for (int i=nops; true; i++) {
-                       // to infinity and beyond ... timewise
-                       resbuf = res;
-                       res = res + cln::recip(cln::expt(i,m_1)) * numeric_harmonic(i, m);
-                       if (cln::zerop(res-resbuf))
-                               break;
-               }
+               // use Hoelder convolution
+               return numeric(zeta_do_Hoelder_convolution(xi, si));
+       }
+
+       return zeta(x, s).hold();
+}
 
-               return numeric(res);
 
+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 mZeta(x1).hold();
+       return zeta(m, s_).hold();
 }
 
-static ex mZeta_series(const ex& x1, const relational& rel, int order, unsigned options)
+
+static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
 {
-       epvector seq;
-       seq.push_back(expair(mZeta(x1), 0));
-       return pseries(rel,seq);
+       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 << ")";
 }
 
-REGISTER_FUNCTION(mZeta, eval_func(mZeta_eval).evalf_func(mZeta_evalf).do_not_evalf_params().series_func(mZeta_series));
+
+unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta").
+                                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