* Removed H(m,s,x)
[ginac.git] / ginac / inifcns_nstdsums.cpp
index 9bff997694da654394944c2b6dbcc9cf2c625203..020ff333b01399329df89dd03e5fc34490087992 100644 (file)
  *     [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 H, Li and zeta is defined according to their order in the
- *      nested sums representation.
+ *    - 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 each argument x_i is smaller than
- *      one. The parameters for every function (n, p or n_i) must be positive integers.
+ *      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 as nested sums. To do it, you have to convert these functions
+ *    - 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. 
  *      
 #include "wildcard.h"
 
 
-//DEBUG
-#include <iostream>
-using namespace std;
-
-
 namespace GiNaC {
 
 
@@ -405,18 +404,6 @@ cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl
 {
        const int j = s.size();
 
-       //DEBUG
-       cout << "m ";
-       for (int i=0; i<s.size(); i++) {
-               cout << s[i] << " ";
-       }
-       cout << endl;
-       cout << "x ";
-       for (int i=0; i<x.size(); i++) {
-               cout << x[i] << " ";
-       }
-       cout << endl;
-
        std::vector<cln::cl_N> t(j);
        cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
 
@@ -438,9 +425,6 @@ cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl
                }
        } while (t[0] != t0buf);
 
-       //DEBUG
-       cout << "end " << q << " " << t[0] << endl;
-       
        return t[0];
 }
 
@@ -457,27 +441,6 @@ cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl
 //////////////////////////////////////////////////////////////////////
 
 
-static ex Li_eval(const ex& x1, const ex& x2)
-{
-       if (x2.is_zero()) {
-               return _ex0;
-       }
-       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));
-               if (is_a<lst>(x2)) {
-                       for (int i=0; i<x2.nops(); i++) {
-                               if (!is_a<numeric>(x2.op(i))) {
-                                       return Li(x1,x2).hold();
-                               }
-                       }
-                       return Li(x1,x2).evalf();
-               }
-               return Li(x1,x2).hold();
-       }
-}
-
-
 static ex Li_evalf(const ex& x1, const ex& x2)
 {
        // classical polylogs
@@ -514,6 +477,27 @@ static ex Li_evalf(const ex& x1, const ex& x2)
 }
 
 
+static ex Li_eval(const ex& x1, const ex& x2)
+{
+       if (x2.is_zero()) {
+               return _ex0;
+       }
+       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));
+               if (is_a<lst>(x2)) {
+                       for (int i=0; i<x2.nops(); i++) {
+                               if (!is_a<numeric>(x2.op(i))) {
+                                       return Li(x1,x2).hold();
+                               }
+                       }
+                       return Li(x1,x2).evalf();
+               }
+               return Li(x1,x2).hold();
+       }
+}
+
+
 static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
 {
        epvector seq;
@@ -536,12 +520,47 @@ static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
 }
 
 
+static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
+{
+       lst m;
+       if (is_a<lst>(m_)) {
+               m = ex_to<lst>(m_);
+       } else {
+               m = lst(m_);
+       }
+       lst x;
+       if (is_a<lst>(x_)) {
+               x = ex_to<lst>(x_);
+       } else {
+               x = lst(x_);
+       }
+       c.s << "\\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,
-               eval_func(Li_eval).
                evalf_func(Li_evalf).
-               do_not_evalf_params().
+               eval_func(Li_eval).
                series_func(Li_series).
-               derivative_func(Li_deriv));
+               derivative_func(Li_deriv).
+               print_func<print_latex>(Li_print_latex).
+               do_not_evalf_params());
 
 
 //////////////////////////////////////////////////////////////////////
@@ -903,22 +922,22 @@ numeric S_num(int n, int p, const numeric& x)
 //////////////////////////////////////////////////////////////////////
 
 
-static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
+static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
 {
-       if (x2 == 1) {
-               return Li(x1+1,x3);
-       }
-       if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) && 
-                       x1.info(info_flags::posint) && x2.info(info_flags::posint)) {
+       if (is_a<numeric>(x1) && is_a<numeric>(x2) && is_a<numeric>(x3)) {
                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();
 }
 
 
-static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
+static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
 {
-       if (is_a<numeric>(x1) && is_a<numeric>(x2) && is_a<numeric>(x3)) {
+       if (x2 == 1) {
+               return Li(x1+1,x3);
+       }
+       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));
        }
        return S(x1,x2,x3).hold();
@@ -947,19 +966,32 @@ static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param
 }
 
 
+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,
-               eval_func(S_eval).
                evalf_func(S_evalf).
-               do_not_evalf_params().
+               eval_func(S_eval).
                series_func(S_series).
-               derivative_func(S_deriv));
+               derivative_func(S_deriv).
+               print_func<print_latex>(S_print_latex).
+               do_not_evalf_params());
 
 
 //////////////////////////////////////////////////////////////////////
 //
-// Harmonic polylogarithm  H(m,x) and H(m,s,x)
+// Harmonic polylogarithm  H(m,x)
 //
-// helper function
+// helper functions
 //
 //////////////////////////////////////////////////////////////////////
 
@@ -968,8 +1000,244 @@ REGISTER_FUNCTION(S,
 namespace {
 
 
-// forward declaration
-ex convert_from_RV(const lst& parameterlst, const ex& arg);
+// 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);
+                       }
+               }
+       }
+       for (; acc > 1; acc--) {
+               throw std::runtime_error("ERROR!");
+               m.append(0);
+       }
+       
+       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& l)
+{
+       symbol xtemp("xtemp");
+       map_trafo_H_reduce_trailing_zeros filter;
+       map_trafo_H_convert_to_zeta filter2;
+       return filter2(filter(H(l, xtemp).hold())).subs(xtemp == 1);
+}
+
+
+// convert signs form Li to H representation
+// not used yet!
+lst convert_parameter_Li_to_H(const lst& l, ex& pf)
+{
+       lst res;
+       lst::const_iterator it = l.begin();
+       ex signum = *it;
+       pf = *it;
+       res.append(*it);
+       it++;
+       while (it != l.end()) {
+               signum = *it * signum;
+               res.append(signum);
+               pf *= signum;
+               it++;
+       }
+
+       return res;
+}
 
 
 // multiplies an one-dimensional H with another H
@@ -1072,9 +1340,9 @@ struct map_trafo_H_mult : public map_function
 };
 
 
-// do integration [ReV] (49)
-// put parameter 1 in front of existing parameters
-ex trafo_H_prepend_one(const ex& e, const ex& arg)
+// 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;
@@ -1095,17 +1363,18 @@ ex trafo_H_prepend_one(const ex& e, const ex& arg)
        }
        if (h != 0) {
                lst newparameter = ex_to<lst>(h.op(0));
-               newparameter.prepend(1);
-               return e.subs(h == H(newparameter, h.op(1)).hold());
+               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(1),1-arg).hold();
+               return e * (-H(lst(0),1/arg).hold());
        }
 }
 
 
 // do integration [ReV] (55)
-// put parameter 0 in front of existing parameters
-ex trafo_H_prepend_zero(const ex& e, const ex& arg)
+// 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;
@@ -1126,70 +1395,171 @@ ex trafo_H_prepend_zero(const ex& e, const ex& arg)
        }
        if (h != 0) {
                lst newparameter = ex_to<lst>(h.op(0));
-               newparameter.prepend(0);
-               ex addzeta = convert_from_RV(newparameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
+               newparameter.prepend(-1);
+               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());
+               ex addzeta = convert_H_to_zeta(lst(-1));
+               return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
        }
 }
 
 
-// do x -> 1-x transformation
-struct map_trafo_H_1mx : public map_function
+// 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 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") {
-
+       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);
 
-                               // if all parameters are either zero or one return the transformed function
-                               if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) {
-                                       lst newparameter;
-                                       for (int i=parameter.nops(); i>0; i--) {
-                                               newparameter.append(0);
+                               // 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;
+                                               }
                                        }
-                                       return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
-                               } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) {
-                                       lst newparameter;
-                                       for (int i=parameter.nops(); i>0; i--) {
-                                               newparameter.append(1);
+                                       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());
                                        }
-                                       return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
                                }
 
                                lst newparameter = parameter;
                                newparameter.remove_first();
 
                                if (parameter.op(0) == 0) {
-
+                                       
                                        // leading zero
-                                       ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
-                                       map_trafo_H_1mx recursion;
+                                       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_prepend_one(buffer.op(i), arg);
+                                                       res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
                                                }
                                        } else {
-                                               res -= trafo_H_prepend_one(buffer, arg);
+                                               res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
                                        }
                                        return res;
 
                                } else {
 
                                        // leading one
-                                       map_trafo_H_1mx recursion;
+                                       map_trafo_H_1overx recursion;
                                        map_trafo_H_mult unify;
-                                       ex res;
+                                       ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
                                        int firstzero = 0;
                                        while (parameter.op(firstzero) == 1) {
                                                firstzero++;
@@ -1206,8 +1576,8 @@ struct map_trafo_H_1mx : public map_function
                                                }
                                                res -= H(newparameter, arg).hold();
                                        }
-                                       return (unify((-H(lst(0), 1-arg).hold() * recursion(H(newparameter, arg).hold())).expand()) +
-                                                       recursion(res)) / firstzero;
+                                       res = recursion(res).expand() / firstzero;
+                                       return unify(res);
 
                                }
 
@@ -1218,8 +1588,8 @@ struct map_trafo_H_1mx : public map_function
 };
 
 
-// do x -> 1/x transformation
-struct map_trafo_H_1overx : public map_function
+// do x -> (1-x)/(1+x) transformation
+struct map_trafo_H_1mxt1px : public map_function
 {
        ex operator()(const ex& e)
        {
@@ -1234,37 +1604,83 @@ struct map_trafo_H_1overx : public map_function
                                lst parameter = ex_to<lst>(e.op(0));
                                ex arg = e.op(1);
 
-                               // if all parameters are either zero or one return the transformed function
-                               if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) {
-                                       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());
-                               } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) {
-                                       return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
+                               // 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_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
-                                       map_trafo_H_1overx recursion;
+                                       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_prepend_zero(buffer.op(i), arg);
+                                                       res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
                                                }
                                        } else {
-                                               res += trafo_H_prepend_zero(buffer, arg);
+                                               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_1overx recursion;
+                                       map_trafo_H_1mxt1px recursion;
                                        map_trafo_H_mult unify;
                                        ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
                                        int firstzero = 0;
@@ -1295,187 +1711,6 @@ struct map_trafo_H_1overx : public map_function
 };
 
 
-// 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();
-                                       for (ex i=0; i<lastentry; i++) {
-                                               if (parameter[i] > 0) {
-                                                       parameter[i]++;
-                                                       result -= (parameter[i]-1) * H(parameter, arg).hold();
-                                                       parameter[i]--;
-                                               } else {
-                                                       parameter[i]--;
-                                                       result -= abs(parameter[i]+1) * H(parameter, arg).hold();
-                                                       parameter[i]++;
-                                               }
-                                       }
-                                       
-                                       if (lastentry < parameter.nops()) {
-                                               result = result / (parameter.nops()-lastentry+1);
-                                               return result.map(*this);
-                                       } else {
-                                               return result;
-                                       }
-                               }
-                       }
-               }
-               return e;
-       }
-};
-
-
-// transform H(m,x) with signed m to H(m,s,x)
-struct map_trafo_H_convert_signed_m : 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);
-                               bool signedflag = false;
-                               for (int i=0; i<parameter.nops(); i++) {
-                                       if (parameter.op(i) < 0) {
-                                               signedflag = true;
-                                               break;
-                                       }
-                               }
-                               if (signedflag) {
-                                       lst signs;
-                                       for (int i=0; i<parameter.nops(); i++) {
-                                               if (parameter.op(i) > 0) {
-                                                       signs.append(1);
-                                               } else {
-                                                       signs.append(-1);
-                                                       parameter.let_op(i) = -parameter.op(i);
-                                               }
-                                       }
-                                       return H(parameter, signs, arg).hold();
-                               }
-                       }
-               }
-               return e;
-       }
-};
-
-
-// recursively call convert_from_RV on expression
-struct map_trafo_H_convert : public map_function
-{
-       ex operator()(const ex& e)
-       {
-               if (is_a<add>(e) || is_a<mul>(e) || is_a<power>(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);
-                               return convert_from_RV(parameter, arg);
-                       }
-               }
-               return e;
-       }
-};
-
-
-// translate notation from nested sums to Remiddi/Vermaseren
-lst convert_to_RV(const lst& o)
-{
-       lst res;
-       for (lst::const_iterator it = o.begin(); it != o.end(); it++) {
-               if (*it > 0) {
-                       for (ex i=0; i<(*it)-1; i++) {
-                               res.append(0);
-                       }
-                       res.append(1);
-               } else {
-                       for (ex i=0; i<(-*it)-1; i++) {
-                               res.append(0);
-                       }
-                       res.append(-1);
-               }
-       }
-       return res;
-}
-
-
-// translate notation from Remiddi/Vermaseren to nested sums
-ex convert_from_RV(const lst& parameterlst, const ex& arg)
-{
-       lst newparameterlst;
-
-       lst::const_iterator it = parameterlst.begin();
-       int count = 1;
-       while (it != parameterlst.end()) {
-               if (*it == 0) {
-                       count++;
-               } else {
-                       newparameterlst.append((*it>0) ? count : -count);
-                       count = 1;
-               }
-               it++;
-       }
-       for (int i=1; i<count; i++) {
-               newparameterlst.append(0);
-       }
-       
-       map_trafo_H_reduce_trailing_zeros filter1;
-       map_trafo_H_convert_signed_m filter2;
-       return filter2(filter1(H(newparameterlst, arg).hold()));
-}
-
-
 // do the actual summation.
 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
 {
@@ -1497,7 +1732,7 @@ cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
                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];
 }
 
@@ -1514,222 +1749,148 @@ cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
 //////////////////////////////////////////////////////////////////////
 
 
-static ex H2_eval(const ex& x1, const ex& x2)
-{
-       if (x2 == 0) {
-               return 0;
-       }
-       if (x2 == 1) {
-               return zeta(x1);
-       }
-       if (x1.nops() == 1) {
-               return Li(x1.op(0), x2);
-       }
-       if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) {
-               return H(x1,x2).evalf();
-       }
-       return H(x1,x2).hold();
-}
-
-
-static ex H2_evalf(const ex& x1, const ex& x2)
+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::posint)) {
+                       if (!x1.op(i).info(info_flags::integer)) {
                                return H(x1,x2).hold();
                        }
                }
                if (x1.nops() < 1) {
-                       return _ex1;
-               }
-               if (x1.nops() == 1) {
-                       return Li(x1.op(0), x2).evalf();
-               }
-               cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
-               if (x == 1) {
-                       return zeta(x1).evalf();
+                       return H(x1,x2).hold();
                }
 
-               // choose trafo
-               if (cln::abs(x) > 1) {
+               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_1overx trafo;
-                       ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
-                       map_trafo_H_convert converter;
-                       res = converter(res);
-                       return res.subs(xtemp==x2).evalf();
+                       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 x->1-x transformation produces a lot of terms, it is only
-               // efficient for argument near one.
-               if (cln::realpart(x) > 0.95) {
-                       symbol xtemp("xtemp");
-                       map_trafo_H_1mx trafo;
-                       ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
-                       map_trafo_H_convert converter;
-                       res = converter(res);
-                       return res.subs(xtemp==x2).evalf();
+               // 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));
+                       }
                }
 
-               // no trafo -> do summation
-               int count = x1.nops();
-               std::vector<int> r(count);
-               for (int i=0; i<count; i++) {
-                       r[i] = ex_to<numeric>(x1.op(i)).to_int();
+               ex res = 1;     
+               
+               // 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;
+                               }
+                       }
                }
 
-               return numeric(H_do_sum(r,x));
-       }
-
-       return H(x1,x2).hold();
-}
-
-
-static ex H2_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
-{
-       epvector seq;
-       seq.push_back(expair(H(x1,x2), 0));
-       return pseries(rel,seq);
-}
-
-
-static ex H2_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
-{
-       GINAC_ASSERT(deriv_param < 2);
-       if (deriv_param == 0) {
-               return _ex0;
-       }
-       if (is_a<lst>(x1)) {
-               lst newparameter = ex_to<lst>(x1);
-               if (x1.op(0) == 1) {
-                       newparameter.remove_first();
-                       return 1/(1-x2) * H(newparameter, x2);
+               // 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 {
-                       newparameter[0]--;
-                       return H(newparameter, x2).hold() / x2;
-               }
-       } else {
-               if (x1 == 1) {
-                       return 1/(1-x2);
-               } else {
-                       return H(x1-1, x2).hold() / x2;
+                       // x -> 1/x
+                       map_trafo_H_1overx trafo;
+                       res *= trafo(H(m, xtemp));
                }
-       }
-}
-
 
-unsigned H2_SERIAL::serial =
-                       function::register_new(function_options("H").
-                                               eval_func(H2_eval).
-                                               evalf_func(H2_evalf).
-                                               do_not_evalf_params().
-                                               derivative_func(H2_deriv).
-                                               latex_name("\\mbox{H}").
-                                               overloaded(2));
+               // 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();
+       }
 
-//////////////////////////////////////////////////////////////////////
-//
-// Harmonic polylogarithm  H(m,s,x)
-//
-// GiNaC function
-//
-//////////////////////////////////////////////////////////////////////
+       return H(x1,x2).hold();
+}
 
 
-static ex H3_eval(const ex& x1, const ex& x2, const ex& x3)
+static ex H_eval(const ex& x1, const ex& x2)
 {
-       if (x3 == 0) {
+       if (x2 == 0) {
                return 0;
        }
-       if (x3 == 1) {
-               return zeta(x1, x2);
-       }
-       if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational))) {
-               return H(x1, x2, x3).evalf();
+//TODO
+//     if (x2 == 1) {
+//             return zeta(x1);
+//     }
+//     if (x1.nops() == 1) {
+//             return Li(x1.op(0), x2);
+//     }
+       if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) {
+               return H(x1,x2).evalf();
        }
-       return H(x1, x2, x3).hold();
-}
-
-
-static ex H3_evalf(const ex& x1, const ex& x2, const ex& x3)
-{
-       if (is_a<lst>(x1) && is_a<numeric>(x3)) {
-               for (int i=0; i<x1.nops(); i++) {
-                       if (!x1.op(i).info(info_flags::posint)) {
-                               return H(x1, x2, x3).hold();
-                       }
-               }
-               if (x1.nops() < 1) {
-                       return _ex1;
-               }
-               if (x1.nops() == 1) {
-                       return x2.op(0) * Li(x1.op(0), x2.op(0)*x3).evalf();
-               }
-               cln::cl_N x = ex_to<numeric>(x3).to_cl_N();
-               if (x == 1) {
-                       return zeta(x1, x2).evalf();
-               }
-
-               // choose trafo
-               if (cln::abs(x) > 1) {
-                       //TODO
-                       return H(x1, x2, x3).hold();
-//                     symbol xtemp("xtemp");
-//                     lst para = ex_to<lst>(x1);
-//                     for (int i=0; i<para.nops(); i++) {
-//                             para.let_op(i) = para.op(i) * x2.op(i);
-//                     }
-//                     map_trafo_H_1overx trafo;
-//                     ex res = trafo(H(convert_to_RV(para), xtemp));
-//                     map_trafo_H_convert converter;
-//                     res = converter(res);
-//                     return res.subs(xtemp==x3).evalf();
-               }
-
-//             // since the x->1-x transformation produces a lot of terms, it is only
-//             // efficient for argument near one.
-//             if (cln::realpart(x) > 0.95) {
-//                     symbol xtemp("xtemp");
-//                     map_trafo_H_1mx trafo;
-//                     ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
-//                     map_trafo_H_convert converter;
-//                     res = converter(res);
-//                     return res.subs(xtemp==x2).evalf();
-//             }
-
-               // no trafo -> do summation
-               int count = x1.nops();
-               std::vector<int> m(count);
-               std::vector<cln::cl_N> s(count);
-               cln::cl_N signbuf = 1;
-               for (int i=0; i<count; i++) {
-                       m[i] = ex_to<numeric>(x1.op(i)).to_int();
-                       signbuf = signbuf * ex_to<numeric>(x2.op(i)).to_cl_N();
-                       s[i] = signbuf;
-               }
-               s[0] = s[0] * ex_to<numeric>(x3).to_cl_N();
-
-               return numeric(signbuf * multipleLi_do_sum(m, s));
-       }
-
-       return H(x1, x2, x3).hold();
+       return H(x1,x2).hold();
 }
 
 
-static ex H3_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options)
+static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
 {
        epvector seq;
-       seq.push_back(expair(H(x1, x2, x3), 0));
-       return pseries(rel, seq);
+       seq.push_back(expair(H(x1,x2), 0));
+       return pseries(rel,seq);
 }
 
 
-static ex H3_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param)
+static ex H_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
 {
-       //TODO
-       
        GINAC_ASSERT(deriv_param < 2);
        if (deriv_param == 0) {
                return _ex0;
@@ -1753,37 +1914,47 @@ static ex H3_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_para
 }
 
 
-unsigned H3_SERIAL::serial =
-                       function::register_new(function_options("H").
-                                               eval_func(H3_eval).
-                                               evalf_func(H3_evalf).
-                                               do_not_evalf_params().
-                                               derivative_func(H3_deriv).
-                                               latex_name("\\mbox{H}").
-                                               overloaded(2));
+static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
+{
+       lst m;
+       if (is_a<lst>(m_)) {
+               m = ex_to<lst>(m_);
+       } else {
+               m = lst(m_);
+       }
+       c.s << "\\mbox{H}_{";
+       lst::const_iterator itm = m.begin();
+       (*itm).print(c);
+       itm++;
+       for (; itm != m.end(); itm++) {
+               c.s << ",";
+               (*itm).print(c);
+       }
+       c.s << "}(";
+       x.print(c);
+       c.s << ")";
+}
+
 
+REGISTER_FUNCTION(H,
+               evalf_func(H_evalf).
+               eval_func(H_eval).
+               series_func(H_series).
+               derivative_func(H_deriv).
+               print_func<print_latex>(H_print_latex).
+               do_not_evalf_params());
 
-ex convert_H_notation(const ex& parameterlst, const ex& arg)
+
+// takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
+ex convert_H_to_Li(const ex& parameterlst, const ex& arg)
 {
+       map_trafo_H_reduce_trailing_zeros filter;
+       map_trafo_H_convert_to_Li filter2;
        if (is_a<lst>(parameterlst)) {
-               for (int i=0; i<parameterlst.nops(); i++) {
-                       if (parameterlst.op(i) == 1) continue;
-                       if (parameterlst.op(i) == 0) continue;
-                       if (parameterlst.op(i) == -1) continue;
-                       throw std::runtime_error("first parameter has to be a list containing only 0, 1 or -1!");
-               }
-               return convert_from_RV(ex_to<lst>(parameterlst), arg).eval();
-       }
-       if (parameterlst == 1) {
-               return -log(1-arg);
-       }
-       if (parameterlst == 0) {
-               return log(arg);
-       }
-       if (parameterlst == -1) {
-               return log(1+arg);
+               return filter2(filter(H(parameterlst, arg).hold())).eval();
+       } else {
+               return filter2(filter(H(lst(parameterlst), arg).hold())).eval();
        }
-       throw std::runtime_error("first parameter has to be a list containing only 0, 1 or -1!");
 }
 
 
@@ -2244,13 +2415,33 @@ static ex zeta1_deriv(const ex& x, unsigned deriv_param)
 }
 
 
+static void zeta1_print_latex(const ex& x, const print_context& c)
+{
+       c.s << "\\zeta(";
+       if (is_a<lst>(x)) {
+               lst arg;
+               arg = ex_to<lst>(x);
+               lst::const_iterator it = arg.begin();
+               (*it).print(c);
+               it++;
+               for (; it != arg.end(); it++) {
+                       c.s << ",";
+                       (*it).print(c);
+               }
+       } else {
+               x.print(c);
+       }
+       c.s << ")";
+}
+
+
 unsigned zeta1_SERIAL::serial =
                        function::register_new(function_options("zeta").
-                                               eval_func(zeta1_eval).
                                                evalf_func(zeta1_evalf).
-                                               do_not_evalf_params().
+                                               eval_func(zeta1_eval).
                                                derivative_func(zeta1_deriv).
-                                               latex_name("\\zeta").
+                                               print_func<print_latex>(zeta1_print_latex).
+                                               do_not_evalf_params().
                                                overloaded(2));
 
 
@@ -2345,13 +2536,53 @@ static ex zeta2_deriv(const ex& x, const ex& s, unsigned deriv_param)
 }
 
 
+static void zeta2_print_latex(const ex& x, const ex& s, const print_context& c)
+{
+       lst arg;
+       if (is_a<lst>(x)) {
+               arg = ex_to<lst>(x);
+       } else {
+               arg = lst(x);
+       }
+       lst sig;
+       if (is_a<lst>(s)) {
+               sig = ex_to<lst>(s);
+       } else {
+               sig = lst(s);
+       }
+       c.s << "\\zeta(";
+       lst::const_iterator itarg = arg.begin();
+       lst::const_iterator itsig = sig.begin();
+       if (*itsig < 0) {
+               c.s << "\\overline{";
+               (*itarg).print(c);
+               c.s << "}";
+       } else {
+               (*itarg).print(c);
+       }
+       itsig++;
+       itarg++;
+       for (; itarg != arg.end(); itarg++, itsig++) {
+               c.s << ",";
+               if (*itsig < 0) {
+                       c.s << "\\overline{";
+                       (*itarg).print(c);
+                       c.s << "}";
+               } else {
+                       (*itarg).print(c);
+               }
+       }
+       c.s << ")";
+}
+
+
 unsigned zeta2_SERIAL::serial =
                        function::register_new(function_options("zeta").
-                                               eval_func(zeta2_eval).
                                                evalf_func(zeta2_evalf).
-                                               do_not_evalf_params().
+                                               eval_func(zeta2_eval).
                                                derivative_func(zeta2_deriv).
-                                               latex_name("\\zeta").
+                                               print_func<print_latex>(zeta2_print_latex).
+                                               do_not_evalf_params().
                                                overloaded(2));