ginac/inifcns_nstdsums.cpp

   1 /** @file inifcns_nstdsums.cpp
   2  *
   3  *  Implementation of some special functions that have a representation as nested sums.
   4  *
   5  *  The functions are:
   6  *    classical polylogarithm              Li(n,x)
   7  *    multiple polylogarithm               Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
   8  *    nielsen's generalized polylogarithm  S(n,p,x)
   9  *    harmonic polylogarithm               H(m,x) or H(lst(m_1,...,m_k),x)
  10  *    multiple zeta value                  zeta(m) or zeta(lst(m_1,...,m_k))
  11  *    alternating Euler sum                zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
  12  *
  13  *  Some remarks:
  14  *
  15  *    - All formulae used can be looked up in the following publications:
  16  *      [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
  17  *      [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
  18  *      [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
  19  *      [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
  20  *
  21  *    - The order of parameters and arguments of Li and zeta is defined according to the nested sums
  22  *      representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
  23  *      0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
  24  *      number --- notation.
  25  *
  26  *    - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
  27  *      the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product
  28  *      x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers.
  29  *      If you want to have an alternating Euler sum, you have to give the signs of the parameters as a
  30  *      second argument s to zeta(m,s) containing 1 and -1.
  31  *
  32  *    - The calculation of classical polylogarithms is speed up by using Bernoulli numbers and
  33  *      look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
  34  *      [Cra] and [BBB] for speed up.
  35  *
  36  *    - The functions have no series expansion into nested sums. To do this, you have to convert these functions
  37  *      into the appropriate objects from the nestedsums library, do the expansion and convert the
  38  *      result back.
  39  *
  40  *    - Numerical testing of this implementation has been performed by doing a comparison of results
  41  *      between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
  42  *      by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
  43  *      comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
  44  *      around |x|=1 along with comparisons to corresponding zeta functions.
  45  *
  46  */
  47
  48 /*
  49  *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
  50  *
  51  *  This program is free software; you can redistribute it and/or modify
  52  *  it under the terms of the GNU General Public License as published by
  53  *  the Free Software Foundation; either version 2 of the License, or
  54  *  (at your option) any later version.
  55  *
  56  *  This program is distributed in the hope that it will be useful,
  57  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  58  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  59  *  GNU General Public License for more details.
  60  *
  61  *  You should have received a copy of the GNU General Public License
  62  *  along with this program; if not, write to the Free Software
  63  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  64  */
  65
  66 #include <stdexcept>
  67 #include <vector>
  68 #include <cln/cln.h>
  69
  70 #include "inifcns.h"
  71
  72 #include "add.h"
  73 #include "constant.h"
  74 #include "lst.h"
  75 #include "mul.h"
  76 #include "numeric.h"
  77 #include "operators.h"
  78 #include "power.h"
  79 #include "pseries.h"
  80 #include "relational.h"
  81 #include "symbol.h"
  82 #include "utils.h"
  83 #include "wildcard.h"
  84
  85
  86 namespace GiNaC {
  87
  88
  89 //////////////////////////////////////////////////////////////////////
  90 //
  91 // Classical polylogarithm  Li(n,x)
  92 //
  93 // helper functions
  94 //
  95 //////////////////////////////////////////////////////////////////////
  96
  97
  98 // anonymous namespace for helper functions
  99 namespace {
 100
 101
 102 // lookup table for factors built from Bernoulli numbers
 103 // see fill_Xn()
 104 std::vector<std::vector<cln::cl_N> > Xn;
 105 int xnsize = 0;
 106
 107
 108 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
 109 // With these numbers the polylogs can be calculated as follows:
 110 //   Li_p (x)  =  \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with  u = -log(1-x)
 111 //   X_0(n) = B_n (Bernoulli numbers)
 112 //   X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
 113 // The calculation of Xn depends on X0 and X{n-1}.
 114 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
 115 // This results in a slightly more complicated algorithm for the X_n.
 116 // The first index in Xn corresponds to the index of the polylog minus 2.
 117 // The second index in Xn corresponds to the index from the actual sum.
 118 void fill_Xn(int n)
 119 {
 120         // rule of thumb. needs to be improved. TODO
 121         const int initsize = Digits * 3 / 2;
 122
 123         if (n>1) {
 124                 // calculate X_2 and higher (corresponding to Li_4 and higher)
 125                 std::vector<cln::cl_N> buf(initsize);
 126                 std::vector<cln::cl_N>::iterator it = buf.begin();
 127                 cln::cl_N result;
 128                 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
 129                 it++;
 130                 for (int i=2; i<=initsize; i++) {
 131                         if (i&1) {
 132                                 result = 0; // k == 0
 133                         } else {
 134                                 result = Xn[0][i/2-1]; // k == 0
 135                         }
 136                         for (int k=1; k<i-1; k++) {
 137                                 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
 138                                         result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
 139                                 }
 140                         }
 141                         result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
 142                         result = result + Xn[n-1][i-1] / (i+1); // k == i
 143
 144                         *it = result;
 145                         it++;
 146                 }
 147                 Xn.push_back(buf);
 148         } else if (n==1) {
 149                 // special case to handle the X_0 correct
 150                 std::vector<cln::cl_N> buf(initsize);
 151                 std::vector<cln::cl_N>::iterator it = buf.begin();
 152                 cln::cl_N result;
 153                 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
 154                 it++;
 155                 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
 156                 it++;
 157                 for (int i=3; i<=initsize; i++) {
 158                         if (i & 1) {
 159                                 result = -Xn[0][(i-3)/2]/2;
 160                                 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
 161                                 it++;
 162                         } else {
 163                                 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
 164                                 for (int k=1; k<i/2; k++) {
 165                                         result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
 166                                 }
 167                                 *it = result;
 168                                 it++;
 169                         }
 170                 }
 171                 Xn.push_back(buf);
 172         } else {
 173                 // calculate X_0
 174                 std::vector<cln::cl_N> buf(initsize/2);
 175                 std::vector<cln::cl_N>::iterator it = buf.begin();
 176                 for (int i=1; i<=initsize/2; i++) {
 177                         *it = bernoulli(i*2).to_cl_N();
 178                         it++;
 179                 }
 180                 Xn.push_back(buf);
 181         }
 182
 183         xnsize++;
 184 }
 185
 186
 187 // calculates Li(2,x) without Xn
 188 cln::cl_N Li2_do_sum(const cln::cl_N& x)
 189 {
 190         cln::cl_N res = x;
 191         cln::cl_N resbuf;
 192         cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
 193         cln::cl_I den = 1; // n^2 = 1
 194         unsigned i = 3;
 195         do {
 196                 resbuf = res;
 197                 num = num * x;
 198                 den = den + i;  // n^2 = 4, 9, 16, ...
 199                 i += 2;
 200                 res = res + num / den;
 201         } while (res != resbuf);
 202         return res;
 203 }
 204
 205
 206 // calculates Li(2,x) with Xn
 207 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
 208 {
 209         std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
 210         cln::cl_N u = -cln::log(1-x);
 211         cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
 212         cln::cl_N res = u - u*u/4;
 213         cln::cl_N resbuf;
 214         unsigned i = 1;
 215         do {
 216                 resbuf = res;
 217                 factor = factor * u*u / (2*i * (2*i+1));
 218                 res = res + (*it) * factor;
 219                 it++; // should we check it? or rely on initsize? ...
 220                 i++;
 221         } while (res != resbuf);
 222         return res;
 223 }
 224
 225
 226 // calculates Li(n,x), n>2 without Xn
 227 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
 228 {
 229         cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
 230         cln::cl_N res = x;
 231         cln::cl_N resbuf;
 232         int i=2;
 233         do {
 234                 resbuf = res;
 235                 factor = factor * x;
 236                 res = res + factor / cln::expt(cln::cl_I(i),n);
 237                 i++;
 238         } while (res != resbuf);
 239         return res;
 240 }
 241
 242
 243 // calculates Li(n,x), n>2 with Xn
 244 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
 245 {
 246         std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
 247         cln::cl_N u = -cln::log(1-x);
 248         cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
 249         cln::cl_N res = u;
 250         cln::cl_N resbuf;
 251         unsigned i=2;
 252         do {
 253                 resbuf = res;
 254                 factor = factor * u / i;
 255                 res = res + (*it) * factor;
 256                 it++; // should we check it? or rely on initsize? ...
 257                 i++;
 258         } while (res != resbuf);
 259         return res;
 260 }
 261
 262
 263 // forward declaration needed by function Li_projection and C below
 264 numeric S_num(int n, int p, const numeric& x);
 265
 266
 267 // helper function for classical polylog Li
 268 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
 269 {
 270         // treat n=2 as special case
 271         if (n == 2) {
 272                 // check if precalculated X0 exists
 273                 if (xnsize == 0) {
 274                         fill_Xn(0);
 275                 }
 276
 277                 if (cln::realpart(x) < 0.5) {
 278                         // choose the faster algorithm
 279                         // the switching point was empirically determined. the optimal point
 280                         // depends on hardware, Digits, ... so an approx value is okay.
 281                         // it solves also the problem with precision due to the u=-log(1-x) transformation
 282                         if (cln::abs(cln::realpart(x)) < 0.25) {
 283
 284                                 return Li2_do_sum(x);
 285                         } else {
 286                                 return Li2_do_sum_Xn(x);
 287                         }
 288                 } else {
 289                         // choose the faster algorithm
 290                         if (cln::abs(cln::realpart(x)) > 0.75) {
 291                                 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
 292                         } else {
 293                                 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
 294                         }
 295                 }
 296         } else {
 297                 // check if precalculated Xn exist
 298                 if (n > xnsize+1) {
 299                         for (int i=xnsize; i<n-1; i++) {
 300                                 fill_Xn(i);
 301                         }
 302                 }
 303
 304                 if (cln::realpart(x) < 0.5) {
 305                         // choose the faster algorithm
 306                         // with n>=12 the "normal" summation always wins against the method with Xn
 307                         if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
 308                                 return Lin_do_sum(n, x);
 309                         } else {
 310                                 return Lin_do_sum_Xn(n, x);
 311                         }
 312                 } else {
 313                         cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
 314                         for (int j=0; j<n-1; j++) {
 315                                 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
 316                                                   * cln::expt(cln::log(x), j) / cln::factorial(j);
 317                         }
 318                         return result;
 319                 }
 320         }
 321 }
 322
 323
 324 // helper function for classical polylog Li
 325 numeric Li_num(int n, const numeric& x)
 326 {
 327         if (n == 1) {
 328                 // just a log
 329                 return -cln::log(1-x.to_cl_N());
 330         }
 331         if (x.is_zero()) {
 332                 return 0;
 333         }
 334         if (x == 1) {
 335                 // [Kol] (2.22)
 336                 return cln::zeta(n);
 337         }
 338         else if (x == -1) {
 339                 // [Kol] (2.22)
 340                 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
 341         }
 342
 343         // what is the desired float format?
 344         // first guess: default format
 345         cln::float_format_t prec = cln::default_float_format;
 346         const cln::cl_N value = x.to_cl_N();
 347         // second guess: the argument's format
 348         if (!x.real().is_rational())
 349                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
 350         else if (!x.imag().is_rational())
 351                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
 352
 353         // [Kol] (5.15)
 354         if (cln::abs(value) > 1) {
 355                 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
 356                 // check if argument is complex. if it is real, the new polylog has to be conjugated.
 357                 if (cln::zerop(cln::imagpart(value))) {
 358                         if (n & 1) {
 359                                 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
 360                         }
 361                         else {
 362                                 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
 363                         }
 364                 }
 365                 else {
 366                         if (n & 1) {
 367                                 result = result + Li_projection(n, cln::recip(value), prec);
 368                         }
 369                         else {
 370                                 result = result - Li_projection(n, cln::recip(value), prec);
 371                         }
 372                 }
 373                 cln::cl_N add;
 374                 for (int j=0; j<n-1; j++) {
 375                         add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
 376                                     * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
 377                 }
 378                 result = result - add;
 379                 return result;
 380         }
 381         else {
 382                 return Li_projection(n, value, prec);
 383         }
 384 }
 385
 386
 387 } // end of anonymous namespace
 388
 389
 390 //////////////////////////////////////////////////////////////////////
 391 //
 392 // Multiple polylogarithm  Li(n,x)
 393 //
 394 // helper function
 395 //
 396 //////////////////////////////////////////////////////////////////////
 397
 398
 399 // anonymous namespace for helper function
 400 namespace {
 401
 402
 403 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
 404 {
 405         const int j = s.size();
 406
 407         std::vector<cln::cl_N> t(j);
 408         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
 409
 410         cln::cl_N t0buf;
 411         int q = 0;
 412         do {
 413                 t0buf = t[0];
 414                 // do it once ...
 415                 q++;
 416                 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
 417                 for (int k=j-2; k>=0; k--) {
 418                         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]);
 419                 }
 420                 // ... and do it again (to avoid premature drop out due to special arguments)
 421                 q++;
 422                 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
 423                 for (int k=j-2; k>=0; k--) {
 424                         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]);
 425                 }
 426         } while (t[0] != t0buf);
 427
 428         return t[0];
 429 }
 430
 431 // forward declaration for Li_eval()
 432 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
 433
 434
 435 } // end of anonymous namespace
 436
 437
 438 //////////////////////////////////////////////////////////////////////
 439 //
 440 // Classical polylogarithm and multiple polylogarithm  Li(n,x)
 441 //
 442 // GiNaC function
 443 //
 444 //////////////////////////////////////////////////////////////////////
 445
 446
 447 static ex Li_evalf(const ex& x1, const ex& x2)
 448 {
 449         // classical polylogs
 450         if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
 451                 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
 452         }
 453         // multiple polylogs
 454         else if (is_a<lst>(x1) && is_a<lst>(x2)) {
 455                 ex conv = 1;
 456                 for (int i=0; i<x1.nops(); i++) {
 457                         if (!x1.op(i).info(info_flags::posint)) {
 458                                 return Li(x1, x2).hold();
 459                         }
 460                         if (!is_a<numeric>(x2.op(i))) {
 461                                 return Li(x1, x2).hold();
 462                         }
 463                         conv *= x2.op(i);
 464                         if (conv >= 1) {
 465                                 return Li(x1, x2).hold();
 466                         }
 467                 }
 468
 469                 std::vector<int> m;
 470                 std::vector<cln::cl_N> x;
 471                 for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
 472                         m.push_back(ex_to<numeric>(x1.op(i)).to_int());
 473                         x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
 474                 }
 475
 476                 return numeric(multipleLi_do_sum(m, x));
 477         }
 478
 479         return Li(x1,x2).hold();
 480 }
 481
 482
 483 static ex Li_eval(const ex& m_, const ex& x_)
 484 {
 485         if (m_.nops() < 2) {
 486                 ex m;
 487                 if (is_a<lst>(m_)) {
 488                         m = m_.op(0);
 489                 } else {
 490                         m = m_;
 491                 }
 492                 ex x;
 493                 if (is_a<lst>(x_)) {
 494                         x = x_.op(0);
 495                 } else {
 496                         x = x_;
 497                 }
 498                 if (x == _ex0) {
 499                         return _ex0;
 500                 }
 501                 if (x == _ex1) {
 502                         return zeta(m);
 503                 }
 504                 if (x == _ex_1) {
 505                         return (pow(2,1-m)-1) * zeta(m);
 506                 }
 507                 if (m == _ex1) {
 508                         return -log(1-x);
 509                 }
 510                 if (m.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
 511                         return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
 512                 }
 513         } else {
 514                 bool ish = true;
 515                 bool iszeta = true;
 516                 bool iszero = false;
 517                 bool doevalf = false;
 518                 bool doevalfveto = true;
 519                 const lst& m = ex_to<lst>(m_);
 520                 const lst& x = ex_to<lst>(x_);
 521                 lst::const_iterator itm = m.begin();
 522                 lst::const_iterator itx = x.begin();
 523                 for (; itm != m.end(); itm++, itx++) {
 524                         if (!(*itm).info(info_flags::posint)) {
 525                                 return Li(m_, x_).hold();
 526                         }
 527                         if ((*itx != _ex1) && (*itx != _ex_1)) {
 528                                 if (itx != x.begin()) {
 529                                         ish = false;
 530                                 }
 531                                 iszeta = false;
 532                         }
 533                         if (*itx == _ex0) {
 534                                 iszero = true;
 535                         }
 536                         if (!(*itx).info(info_flags::numeric)) {
 537                                 doevalfveto = false;
 538                         }
 539                         if (!(*itx).info(info_flags::crational)) {
 540                                 doevalf = true;
 541                         }
 542                 }
 543                 if (iszeta) {
 544                         return zeta(m_, x_);
 545                 }
 546                 if (iszero) {
 547                         return _ex0;
 548                 }
 549                 if (ish) {
 550                         ex pf;
 551                         lst newm = convert_parameter_Li_to_H(m, x, pf);
 552                         return pf * H(newm, x[0]);
 553                 }
 554                 if (doevalfveto && doevalf) {
 555                         return Li(m_, x_).evalf();
 556                 }
 557         }
 558         return Li(m_, x_).hold();
 559 }
 560
 561
 562 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
 563 {
 564         epvector seq;
 565         seq.push_back(expair(Li(m, x), 0));
 566         return pseries(rel, seq);
 567 }
 568
 569
 570 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
 571 {
 572         GINAC_ASSERT(deriv_param < 2);
 573         if (deriv_param == 0) {
 574                 return _ex0;
 575         }
 576         if (m_.nops() > 1) {
 577                 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
 578         }
 579         ex m;
 580         if (is_a<lst>(m_)) {
 581                 m = m_.op(0);
 582         } else {
 583                 m = m_;
 584         }
 585         ex x;
 586         if (is_a<lst>(x_)) {
 587                 x = x_.op(0);
 588         } else {
 589                 x = x_;
 590         }
 591         if (m > 0) {
 592                 return Li(m-1, x) / x;
 593         } else {
 594                 return 1/(1-x);
 595         }
 596 }
 597
 598
 599 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
 600 {
 601         lst m;
 602         if (is_a<lst>(m_)) {
 603                 m = ex_to<lst>(m_);
 604         } else {
 605                 m = lst(m_);
 606         }
 607         lst x;
 608         if (is_a<lst>(x_)) {
 609                 x = ex_to<lst>(x_);
 610         } else {
 611                 x = lst(x_);
 612         }
 613         c.s << "\\mbox{Li}_{";
 614         lst::const_iterator itm = m.begin();
 615         (*itm).print(c);
 616         itm++;
 617         for (; itm != m.end(); itm++) {
 618                 c.s << ",";
 619                 (*itm).print(c);
 620         }
 621         c.s << "}(";
 622         lst::const_iterator itx = x.begin();
 623         (*itx).print(c);
 624         itx++;
 625         for (; itx != x.end(); itx++) {
 626                 c.s << ",";
 627                 (*itx).print(c);
 628         }
 629         c.s << ")";
 630 }
 631
 632
 633 REGISTER_FUNCTION(Li,
 634                   evalf_func(Li_evalf).
 635                   eval_func(Li_eval).
 636                   series_func(Li_series).
 637                   derivative_func(Li_deriv).
 638                   print_func<print_latex>(Li_print_latex).
 639                   do_not_evalf_params());
 640
 641
 642 //////////////////////////////////////////////////////////////////////
 643 //
 644 // Nielsen's generalized polylogarithm  S(n,p,x)
 645 //
 646 // helper functions
 647 //
 648 //////////////////////////////////////////////////////////////////////
 649
 650
 651 // anonymous namespace for helper functions
 652 namespace {
 653
 654
 655 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
 656 // see fill_Yn()
 657 std::vector<std::vector<cln::cl_N> > Yn;
 658 int ynsize = 0; // number of Yn[]
 659 int ynlength = 100; // initial length of all Yn[i]
 660
 661
 662 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
 663 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
 664 // representing S_{n,p}(x).
 665 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
 666 // equivalent Z-sum.
 667 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
 668 // representing S_{n,p}(x).
 669 // The calculation of Y_n uses the values from Y_{n-1}.
 670 void fill_Yn(int n, const cln::float_format_t& prec)
 671 {
 672         const int initsize = ynlength;
 673         //const int initsize = initsize_Yn;
 674         cln::cl_N one = cln::cl_float(1, prec);
 675
 676         if (n) {
 677                 std::vector<cln::cl_N> buf(initsize);
 678                 std::vector<cln::cl_N>::iterator it = buf.begin();
 679                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
 680                 *it = (*itprev) / cln::cl_N(n+1) * one;
 681                 it++;
 682                 itprev++;
 683                 // sums with an index smaller than the depth are zero and need not to be calculated.
 684                 // calculation starts with depth, which is n+2)
 685                 for (int i=n+2; i<=initsize+n; i++) {
 686                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
 687                         it++;
 688                         itprev++;
 689                 }
 690                 Yn.push_back(buf);
 691         } else {
 692                 std::vector<cln::cl_N> buf(initsize);
 693                 std::vector<cln::cl_N>::iterator it = buf.begin();
 694                 *it = 1 * one;
 695                 it++;
 696                 for (int i=2; i<=initsize; i++) {
 697                         *it = *(it-1) + 1 / cln::cl_N(i) * one;
 698                         it++;
 699                 }
 700                 Yn.push_back(buf);
 701         }
 702         ynsize++;
 703 }
 704
 705
 706 // make Yn longer ...
 707 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
 708 {
 709
 710         cln::cl_N one = cln::cl_float(1, prec);
 711
 712         Yn[0].resize(newsize);
 713         std::vector<cln::cl_N>::iterator it = Yn[0].begin();
 714         it += ynlength;
 715         for (int i=ynlength+1; i<=newsize; i++) {
 716                 *it = *(it-1) + 1 / cln::cl_N(i) * one;
 717                 it++;
 718         }
 719
 720         for (int n=1; n<ynsize; n++) {
 721                 Yn[n].resize(newsize);
 722                 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
 723                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
 724                 it += ynlength;
 725                 itprev += ynlength;
 726                 for (int i=ynlength+n+1; i<=newsize+n; i++) {
 727                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
 728                         it++;
 729                         itprev++;
 730                 }
 731         }
 732
 733         ynlength = newsize;
 734 }
 735
 736
 737 // helper function for S(n,p,x)
 738 // [Kol] (7.2)
 739 cln::cl_N C(int n, int p)
 740 {
 741         cln::cl_N result;
 742
 743         for (int k=0; k<p; k++) {
 744                 for (int j=0; j<=(n+k-1)/2; j++) {
 745                         if (k == 0) {
 746                                 if (n & 1) {
 747                                         if (j & 1) {
 748                                                 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
 749                                         }
 750                                         else {
 751                                                 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
 752                                         }
 753                                 }
 754                         }
 755                         else {
 756                                 if (k & 1) {
 757                                         if (j & 1) {
 758                                                 result = result + cln::factorial(n+k-1)
 759                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
 760                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
 761                                         }
 762                                         else {
 763                                                 result = result - cln::factorial(n+k-1)
 764                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
 765                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
 766                                         }
 767                                 }
 768                                 else {
 769                                         if (j & 1) {
 770                                                 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()
 771                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
 772                                         }
 773                                         else {
 774                                                 result = result + cln::factorial(n+k-1)
 775                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
 776                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
 777                                         }
 778                                 }
 779                         }
 780                 }
 781         }
 782         int np = n+p;
 783         if ((np-1) & 1) {
 784                 if (((np)/2+n) & 1) {
 785                         result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
 786                 }
 787                 else {
 788                         result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
 789                 }
 790         }
 791
 792         return result;
 793 }
 794
 795
 796 // helper function for S(n,p,x)
 797 // [Kol] remark to (9.1)
 798 cln::cl_N a_k(int k)
 799 {
 800         cln::cl_N result;
 801
 802         if (k == 0) {
 803                 return 1;
 804         }
 805
 806         result = result;
 807         for (int m=2; m<=k; m++) {
 808                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
 809         }
 810
 811         return -result / k;
 812 }
 813
 814
 815 // helper function for S(n,p,x)
 816 // [Kol] remark to (9.1)
 817 cln::cl_N b_k(int k)
 818 {
 819         cln::cl_N result;
 820
 821         if (k == 0) {
 822                 return 1;
 823         }
 824
 825         result = result;
 826         for (int m=2; m<=k; m++) {
 827                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
 828         }
 829
 830         return result / k;
 831 }
 832
 833
 834 // helper function for S(n,p,x)
 835 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
 836 {
 837         if (p==1) {
 838                 return Li_projection(n+1, x, prec);
 839         }
 840
 841         // check if precalculated values are sufficient
 842         if (p > ynsize+1) {
 843                 for (int i=ynsize; i<p-1; i++) {
 844                         fill_Yn(i, prec);
 845                 }
 846         }
 847
 848         // should be done otherwise
 849         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
 850         cln::cl_N xf = x * one;
 851         //cln::cl_N xf = x * cln::cl_float(1, prec);
 852
 853         cln::cl_N res;
 854         cln::cl_N resbuf;
 855         cln::cl_N factor = cln::expt(xf, p);
 856         int i = p;
 857         do {
 858                 resbuf = res;
 859                 if (i-p >= ynlength) {
 860                         // make Yn longer
 861                         make_Yn_longer(ynlength*2, prec);
 862                 }
 863                 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? ...
 864                 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
 865                 factor = factor * xf;
 866                 i++;
 867         } while (res != resbuf);
 868
 869         return res;
 870 }
 871
 872
 873 // helper function for S(n,p,x)
 874 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
 875 {
 876         // [Kol] (5.3)
 877         if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
 878
 879                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
 880                                    * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
 881
 882                 for (int s=0; s<n; s++) {
 883                         cln::cl_N res2;
 884                         for (int r=0; r<p; r++) {
 885                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
 886                                               * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
 887                         }
 888                         result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
 889                 }
 890
 891                 return result;
 892         }
 893
 894         return S_do_sum(n, p, x, prec);
 895 }
 896
 897
 898 // helper function for S(n,p,x)
 899 numeric S_num(int n, int p, const numeric& x)
 900 {
 901         if (x == 1) {
 902                 if (n == 1) {
 903                     // [Kol] (2.22) with (2.21)
 904                         return cln::zeta(p+1);
 905                 }
 906
 907                 if (p == 1) {
 908                     // [Kol] (2.22)
 909                         return cln::zeta(n+1);
 910                 }
 911
 912                 // [Kol] (9.1)
 913                 cln::cl_N result;
 914                 for (int nu=0; nu<n; nu++) {
 915                         for (int rho=0; rho<=p; rho++) {
 916                                 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
 917                                                   * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
 918                         }
 919                 }
 920                 result = result * cln::expt(cln::cl_I(-1),n+p-1);
 921
 922                 return result;
 923         }
 924         else if (x == -1) {
 925                 // [Kol] (2.22)
 926                 if (p == 1) {
 927                         return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
 928                 }
 929 //              throw std::runtime_error("don't know how to evaluate this function!");
 930         }
 931
 932         // what is the desired float format?
 933         // first guess: default format
 934         cln::float_format_t prec = cln::default_float_format;
 935         const cln::cl_N value = x.to_cl_N();
 936         // second guess: the argument's format
 937         if (!x.real().is_rational())
 938                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
 939         else if (!x.imag().is_rational())
 940                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
 941
 942
 943         // [Kol] (5.3)
 944         if (cln::realpart(value) < -0.5) {
 945
 946                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
 947                                    * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
 948
 949                 for (int s=0; s<n; s++) {
 950                         cln::cl_N res2;
 951                         for (int r=0; r<p; r++) {
 952                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
 953                                               * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
 954                         }
 955                         result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
 956                 }
 957
 958                 return result;
 959
 960         }
 961         // [Kol] (5.12)
 962         if (cln::abs(value) > 1) {
 963
 964                 cln::cl_N result;
 965
 966                 for (int s=0; s<p; s++) {
 967                         for (int r=0; r<=s; r++) {
 968                                 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
 969                                                   / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
 970                                                   * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
 971                         }
 972                 }
 973                 result = result * cln::expt(cln::cl_I(-1),n);
 974
 975                 cln::cl_N res2;
 976                 for (int r=0; r<n; r++) {
 977                         res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
 978                 }
 979                 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
 980
 981                 result = result + cln::expt(cln::cl_I(-1),p) * res2;
 982
 983                 return result;
 984         }
 985         else {
 986                 return S_projection(n, p, value, prec);
 987         }
 988 }
 989
 990
 991 } // end of anonymous namespace
 992
 993
 994 //////////////////////////////////////////////////////////////////////
 995 //
 996 // Nielsen's generalized polylogarithm  S(n,p,x)
 997 //
 998 // GiNaC function
 999 //
1000 //////////////////////////////////////////////////////////////////////
1001
1002
1003 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1004 {
1005         if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a<numeric>(x)) {
1006                 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1007         }
1008         return S(n, p, x).hold();
1009 }
1010
1011
1012 static ex S_eval(const ex& n, const ex& p, const ex& x)
1013 {
1014         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1015                 if (x == 0) {
1016                         return _ex0;
1017                 }
1018                 if (x == 1) {
1019                         lst m(n+1);
1020                         for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
1021                                 m.append(1);
1022                         }
1023                         return zeta(m);
1024                 }
1025                 if (p == 1) {
1026                         return Li(n+1, x);
1027                 }
1028                 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
1029                         return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1030                 }
1031         }
1032         return S(n, p, x).hold();
1033 }
1034
1035
1036 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
1037 {
1038         epvector seq;
1039         seq.push_back(expair(S(n, p, x), 0));
1040         return pseries(rel, seq);
1041 }
1042
1043
1044 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
1045 {
1046         GINAC_ASSERT(deriv_param < 3);
1047         if (deriv_param < 2) {
1048                 return _ex0;
1049         }
1050         if (n > 0) {
1051                 return S(n-1, p, x) / x;
1052         } else {
1053                 return S(n, p-1, x) / (1-x);
1054         }
1055 }
1056
1057
1058 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
1059 {
1060         c.s << "\\mbox{S}_{";
1061         n.print(c);
1062         c.s << ",";
1063         p.print(c);
1064         c.s << "}(";
1065         x.print(c);
1066         c.s << ")";
1067 }
1068
1069
1070 REGISTER_FUNCTION(S,
1071                   evalf_func(S_evalf).
1072                   eval_func(S_eval).
1073                   series_func(S_series).
1074                   derivative_func(S_deriv).
1075                   print_func<print_latex>(S_print_latex).
1076                   do_not_evalf_params());
1077
1078
1079 //////////////////////////////////////////////////////////////////////
1080 //
1081 // Harmonic polylogarithm  H(m,x)
1082 //
1083 // helper functions
1084 //
1085 //////////////////////////////////////////////////////////////////////
1086
1087
1088 // anonymous namespace for helper functions
1089 namespace {
1090
1091
1092 // convert parameters from H to Li representation
1093 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
1094 // returns true if some parameters are negative
1095 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
1096 {
1097         // expand parameter list
1098         lst mexp;
1099         for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
1100                 if (*it > 1) {
1101                         for (ex count=*it-1; count > 0; count--) {
1102                                 mexp.append(0);
1103                         }
1104                         mexp.append(1);
1105                 } else if (*it < -1) {
1106                         for (ex count=*it+1; count < 0; count++) {
1107                                 mexp.append(0);
1108                         }
1109                         mexp.append(-1);
1110                 } else {
1111                         mexp.append(*it);
1112                 }
1113         }
1114
1115         ex signum = 1;
1116         pf = 1;
1117         bool has_negative_parameters = false;
1118         ex acc = 1;
1119         for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
1120                 if (*it == 0) {
1121                         acc++;
1122                         continue;
1123                 }
1124                 if (*it > 0) {
1125                         m.append((*it+acc-1) * signum);
1126                 } else {
1127                         m.append((*it-acc+1) * signum);
1128                 }
1129                 acc = 1;
1130                 signum = *it;
1131                 pf *= *it;
1132                 if (pf < 0) {
1133                         has_negative_parameters = true;
1134                 }
1135         }
1136         if (has_negative_parameters) {
1137                 for (int i=0; i<m.nops(); i++) {
1138                         if (m.op(i) < 0) {
1139                                 m.let_op(i) = -m.op(i);
1140                                 s.append(-1);
1141                         } else {
1142                                 s.append(1);
1143                         }
1144                 }
1145         }
1146
1147         return has_negative_parameters;
1148 }
1149
1150
1151 // recursivly transforms H to corresponding multiple polylogarithms
1152 struct map_trafo_H_convert_to_Li : public map_function
1153 {
1154         ex operator()(const ex& e)
1155         {
1156                 if (is_a<add>(e) || is_a<mul>(e)) {
1157                         return e.map(*this);
1158                 }
1159                 if (is_a<function>(e)) {
1160                         std::string name = ex_to<function>(e).get_name();
1161                         if (name == "H") {
1162                                 lst parameter;
1163                                 if (is_a<lst>(e.op(0))) {
1164                                                 parameter = ex_to<lst>(e.op(0));
1165                                 } else {
1166                                         parameter = lst(e.op(0));
1167                                 }
1168                                 ex arg = e.op(1);
1169
1170                                 lst m;
1171                                 lst s;
1172                                 ex pf;
1173                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1174                                         s.let_op(0) = s.op(0) * arg;
1175                                         return pf * Li(m, s).hold();
1176                                 } else {
1177                                         for (int i=0; i<m.nops(); i++) {
1178                                                 s.append(1);
1179                                         }
1180                                         s.let_op(0) = s.op(0) * arg;
1181                                         return Li(m, s).hold();
1182                                 }
1183                         }
1184                 }
1185                 return e;
1186         }
1187 };
1188
1189
1190 // recursivly transforms H to corresponding zetas
1191 struct map_trafo_H_convert_to_zeta : public map_function
1192 {
1193         ex operator()(const ex& e)
1194         {
1195                 if (is_a<add>(e) || is_a<mul>(e)) {
1196                         return e.map(*this);
1197                 }
1198                 if (is_a<function>(e)) {
1199                         std::string name = ex_to<function>(e).get_name();
1200                         if (name == "H") {
1201                                 lst parameter;
1202                                 if (is_a<lst>(e.op(0))) {
1203                                                 parameter = ex_to<lst>(e.op(0));
1204                                 } else {
1205                                         parameter = lst(e.op(0));
1206                                 }
1207
1208                                 lst m;
1209                                 lst s;
1210                                 ex pf;
1211                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1212                                         return pf * zeta(m, s);
1213                                 } else {
1214                                         return zeta(m);
1215                                 }
1216                         }
1217                 }
1218                 return e;
1219         }
1220 };
1221
1222
1223 // remove trailing zeros from H-parameters
1224 struct map_trafo_H_reduce_trailing_zeros : public map_function
1225 {
1226         ex operator()(const ex& e)
1227         {
1228                 if (is_a<add>(e) || is_a<mul>(e)) {
1229                         return e.map(*this);
1230                 }
1231                 if (is_a<function>(e)) {
1232                         std::string name = ex_to<function>(e).get_name();
1233                         if (name == "H") {
1234                                 lst parameter;
1235                                 if (is_a<lst>(e.op(0))) {
1236                                         parameter = ex_to<lst>(e.op(0));
1237                                 } else {
1238                                         parameter = lst(e.op(0));
1239                                 }
1240                                 ex arg = e.op(1);
1241                                 if (parameter.op(parameter.nops()-1) == 0) {
1242
1243                                         //
1244                                         if (parameter.nops() == 1) {
1245                                                 return log(arg);
1246                                         }
1247
1248                                         //
1249                                         lst::const_iterator it = parameter.begin();
1250                                         while ((it != parameter.end()) && (*it == 0)) {
1251                                                 it++;
1252                                         }
1253                                         if (it == parameter.end()) {
1254                                                 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1255                                         }
1256
1257                                         //
1258                                         parameter.remove_last();
1259                                         int lastentry = parameter.nops();
1260                                         while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1261                                                 lastentry--;
1262                                         }
1263
1264                                         //
1265                                         ex result = log(arg) * H(parameter,arg).hold();
1266                                         ex acc = 0;
1267                                         for (ex i=0; i<lastentry; i++) {
1268                                                 if (parameter[i] > 0) {
1269                                                         parameter[i]++;
1270                                                         result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
1271                                                         parameter[i]--;
1272                                                         acc = 0;
1273                                                 } else if (parameter[i] < 0) {
1274                                                         parameter[i]--;
1275                                                         result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
1276                                                         parameter[i]++;
1277                                                         acc = 0;
1278                                                 } else {
1279                                                         acc++;
1280                                                 }
1281                                         }
1282
1283                                         if (lastentry < parameter.nops()) {
1284                                                 result = result / (parameter.nops()-lastentry+1);
1285                                                 return result.map(*this);
1286                                         } else {
1287                                                 return result;
1288                                         }
1289                                 }
1290                         }
1291                 }
1292                 return e;
1293         }
1294 };
1295
1296
1297 // returns an expression with zeta functions corresponding to the parameter list for H
1298 ex convert_H_to_zeta(const lst& m)
1299 {
1300         symbol xtemp("xtemp");
1301         map_trafo_H_reduce_trailing_zeros filter;
1302         map_trafo_H_convert_to_zeta filter2;
1303         return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
1304 }
1305
1306
1307 // convert signs form Li to H representation
1308 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
1309 {
1310         lst res;
1311         lst::const_iterator itm = m.begin();
1312         lst::const_iterator itx = ++x.begin();
1313         ex signum = _ex1;
1314         pf = _ex1;
1315         res.append(*itm);
1316         itm++;
1317         while (itx != x.end()) {
1318                 signum *= *itx;
1319                 pf *= signum;
1320                 res.append((*itm) * signum);
1321                 itm++;
1322                 itx++;
1323         }
1324         return res;
1325 }
1326
1327
1328 // multiplies an one-dimensional H with another H
1329 // [ReV] (18)
1330 ex trafo_H_mult(const ex& h1, const ex& h2)
1331 {
1332         ex res;
1333         ex hshort;
1334         lst hlong;
1335         ex h1nops = h1.op(0).nops();
1336         ex h2nops = h2.op(0).nops();
1337         if (h1nops > 1) {
1338                 hshort = h2.op(0).op(0);
1339                 hlong = ex_to<lst>(h1.op(0));
1340         } else {
1341                 hshort = h1.op(0).op(0);
1342                 if (h2nops > 1) {
1343                         hlong = ex_to<lst>(h2.op(0));
1344                 } else {
1345                         hlong = h2.op(0).op(0);
1346                 }
1347         }
1348         for (int i=0; i<=hlong.nops(); i++) {
1349                 lst newparameter;
1350                 int j=0;
1351                 for (; j<i; j++) {
1352                         newparameter.append(hlong[j]);
1353                 }
1354                 newparameter.append(hshort);
1355                 for (; j<hlong.nops(); j++) {
1356                         newparameter.append(hlong[j]);
1357                 }
1358                 res += H(newparameter, h1.op(1)).hold();
1359         }
1360         return res;
1361 }
1362
1363
1364 // applies trafo_H_mult recursively on expressions
1365 struct map_trafo_H_mult : public map_function
1366 {
1367         ex operator()(const ex& e)
1368         {
1369                 if (is_a<add>(e)) {
1370                         return e.map(*this);
1371                 }
1372
1373                 if (is_a<mul>(e)) {
1374
1375                         ex result = 1;
1376                         ex firstH;
1377                         lst Hlst;
1378                         for (int pos=0; pos<e.nops(); pos++) {
1379                                 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1380                                         std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1381                                         if (name == "H") {
1382                                                 for (ex i=0; i<e.op(pos).op(1); i++) {
1383                                                         Hlst.append(e.op(pos).op(0));
1384                                                 }
1385                                                 continue;
1386                                         }
1387                                 } else if (is_a<function>(e.op(pos))) {
1388                                         std::string name = ex_to<function>(e.op(pos)).get_name();
1389                                         if (name == "H") {
1390                                                 if (e.op(pos).op(0).nops() > 1) {
1391                                                         firstH = e.op(pos);
1392                                                 } else {
1393                                                         Hlst.append(e.op(pos));
1394                                                 }
1395                                                 continue;
1396                                         }
1397                                 }
1398                                 result *= e.op(pos);
1399                         }
1400                         if (firstH == 0) {
1401                                 if (Hlst.nops() > 0) {
1402                                         firstH = Hlst[Hlst.nops()-1];
1403                                         Hlst.remove_last();
1404                                 } else {
1405                                         return e;
1406                                 }
1407                         }
1408
1409                         if (Hlst.nops() > 0) {
1410                                 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1411                                 result *= buffer;
1412                                 for (int i=1; i<Hlst.nops(); i++) {
1413                                         result *= Hlst.op(i);
1414                                 }
1415                                 result = result.expand();
1416                                 map_trafo_H_mult recursion;
1417                                 return recursion(result);
1418                         } else {
1419                                 return e;
1420                         }
1421
1422                 }
1423                 return e;
1424         }
1425 };
1426
1427
1428 // do integration [ReV] (55)
1429 // put parameter 0 in front of existing parameters
1430 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
1431 {
1432         ex h;
1433         std::string name;
1434         if (is_a<function>(e)) {
1435                 name = ex_to<function>(e).get_name();
1436         }
1437         if (name == "H") {
1438                 h = e;
1439         } else {
1440                 for (int i=0; i<e.nops(); i++) {
1441                         if (is_a<function>(e.op(i))) {
1442                                 std::string name = ex_to<function>(e.op(i)).get_name();
1443                                 if (name == "H") {
1444                                         h = e.op(i);
1445                                 }
1446                         }
1447                 }
1448         }
1449         if (h != 0) {
1450                 lst newparameter = ex_to<lst>(h.op(0));
1451                 newparameter.prepend(0);
1452                 ex addzeta = convert_H_to_zeta(newparameter);
1453                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1454         } else {
1455                 return e * (-H(lst(0),1/arg).hold());
1456         }
1457 }
1458
1459
1460 // do integration [ReV] (55)
1461 // put parameter -1 in front of existing parameters
1462 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
1463 {
1464         ex h;
1465         std::string name;
1466         if (is_a<function>(e)) {
1467                 name = ex_to<function>(e).get_name();
1468         }
1469         if (name == "H") {
1470                 h = e;
1471         } else {
1472                 for (int i=0; i<e.nops(); i++) {
1473                         if (is_a<function>(e.op(i))) {
1474                                 std::string name = ex_to<function>(e.op(i)).get_name();
1475                                 if (name == "H") {
1476                                         h = e.op(i);
1477                                 }
1478                         }
1479                 }
1480         }
1481         if (h != 0) {
1482                 lst newparameter = ex_to<lst>(h.op(0));
1483                 newparameter.prepend(-1);
1484                 ex addzeta = convert_H_to_zeta(newparameter);
1485                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1486         } else {
1487                 ex addzeta = convert_H_to_zeta(lst(-1));
1488                 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
1489         }
1490 }
1491
1492
1493 // do integration [ReV] (55)
1494 // put parameter -1 in front of existing parameters
1495 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
1496 {
1497         ex h;
1498         std::string name;
1499         if (is_a<function>(e)) {
1500                 name = ex_to<function>(e).get_name();
1501         }
1502         if (name == "H") {
1503                 h = e;
1504         } else {
1505                 for (int i=0; i<e.nops(); i++) {
1506                         if (is_a<function>(e.op(i))) {
1507                                 std::string name = ex_to<function>(e.op(i)).get_name();
1508                                 if (name == "H") {
1509                                         h = e.op(i);
1510                                 }
1511                         }
1512                 }
1513         }
1514         if (h != 0) {
1515                 lst newparameter = ex_to<lst>(h.op(0));
1516                 newparameter.prepend(-1);
1517                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1518         } else {
1519                 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
1520         }
1521 }
1522
1523
1524 // do integration [ReV] (55)
1525 // put parameter 1 in front of existing parameters
1526 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
1527 {
1528         ex h;
1529         std::string name;
1530         if (is_a<function>(e)) {
1531                 name = ex_to<function>(e).get_name();
1532         }
1533         if (name == "H") {
1534                 h = e;
1535         } else {
1536                 for (int i=0; i<e.nops(); i++) {
1537                         if (is_a<function>(e.op(i))) {
1538                                 std::string name = ex_to<function>(e.op(i)).get_name();
1539                                 if (name == "H") {
1540                                         h = e.op(i);
1541                                 }
1542                         }
1543                 }
1544         }
1545         if (h != 0) {
1546                 lst newparameter = ex_to<lst>(h.op(0));
1547                 newparameter.prepend(1);
1548                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1549         } else {
1550                 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
1551         }
1552 }
1553
1554
1555 // do x -> 1/x transformation
1556 struct map_trafo_H_1overx : public map_function
1557 {
1558         ex operator()(const ex& e)
1559         {
1560                 if (is_a<add>(e) || is_a<mul>(e)) {
1561                         return e.map(*this);
1562                 }
1563
1564                 if (is_a<function>(e)) {
1565                         std::string name = ex_to<function>(e).get_name();
1566                         if (name == "H") {
1567
1568                                 lst parameter = ex_to<lst>(e.op(0));
1569                                 ex arg = e.op(1);
1570
1571                                 // special cases if all parameters are either 0, 1 or -1
1572                                 bool allthesame = true;
1573                                 if (parameter.op(0) == 0) {
1574                                         for (int i=1; i<parameter.nops(); i++) {
1575                                                 if (parameter.op(i) != 0) {
1576                                                         allthesame = false;
1577                                                         break;
1578                                                 }
1579                                         }
1580                                         if (allthesame) {
1581                                                 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1582                                         }
1583                                 } else if (parameter.op(0) == -1) {
1584                                         for (int i=1; i<parameter.nops(); i++) {
1585                                                 if (parameter.op(i) != -1) {
1586                                                         allthesame = false;
1587                                                         break;
1588                                                 }
1589                                         }
1590                                         if (allthesame) {
1591                                                 map_trafo_H_mult unify;
1592                                                 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
1593                                                        / factorial(parameter.nops())).expand());
1594                                         }
1595                                 } else {
1596                                         for (int i=1; i<parameter.nops(); i++) {
1597                                                 if (parameter.op(i) != 1) {
1598                                                         allthesame = false;
1599                                                         break;
1600                                                 }
1601                                         }
1602                                         if (allthesame) {
1603                                                 map_trafo_H_mult unify;
1604                                                 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, parameter.nops())
1605                                                        / factorial(parameter.nops())).expand());
1606                                         }
1607                                 }
1608
1609                                 lst newparameter = parameter;
1610                                 newparameter.remove_first();
1611
1612                                 if (parameter.op(0) == 0) {
1613
1614                                         // leading zero
1615                                         ex res = convert_H_to_zeta(parameter);
1616                                         map_trafo_H_1overx recursion;
1617                                         ex buffer = recursion(H(newparameter, arg).hold());
1618                                         if (is_a<add>(buffer)) {
1619                                                 for (int i=0; i<buffer.nops(); i++) {
1620                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
1621                                                 }
1622                                         } else {
1623                                                 res += trafo_H_1tx_prepend_zero(buffer, arg);
1624                                         }
1625                                         return res;
1626
1627                                 } else if (parameter.op(0) == -1) {
1628
1629                                         // leading negative one
1630                                         ex res = convert_H_to_zeta(parameter);
1631                                         map_trafo_H_1overx recursion;
1632                                         ex buffer = recursion(H(newparameter, arg).hold());
1633                                         if (is_a<add>(buffer)) {
1634                                                 for (int i=0; i<buffer.nops(); i++) {
1635                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
1636                                                 }
1637                                         } else {
1638                                                 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
1639                                         }
1640                                         return res;
1641
1642                                 } else {
1643
1644                                         // leading one
1645                                         map_trafo_H_1overx recursion;
1646                                         map_trafo_H_mult unify;
1647                                         ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1648                                         int firstzero = 0;
1649                                         while (parameter.op(firstzero) == 1) {
1650                                                 firstzero++;
1651                                         }
1652                                         for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1653                                                 lst newparameter;
1654                                                 int j=0;
1655                                                 for (; j<=i; j++) {
1656                                                         newparameter.append(parameter[j+1]);
1657                                                 }
1658                                                 newparameter.append(1);
1659                                                 for (; j<parameter.nops()-1; j++) {
1660                                                         newparameter.append(parameter[j+1]);
1661                                                 }
1662                                                 res -= H(newparameter, arg).hold();
1663                                         }
1664                                         res = recursion(res).expand() / firstzero;
1665                                         return unify(res);
1666
1667                                 }
1668
1669                         }
1670                 }
1671                 return e;
1672         }
1673 };
1674
1675
1676 // do x -> (1-x)/(1+x) transformation
1677 struct map_trafo_H_1mxt1px : public map_function
1678 {
1679         ex operator()(const ex& e)
1680         {
1681                 if (is_a<add>(e) || is_a<mul>(e)) {
1682                         return e.map(*this);
1683                 }
1684
1685                 if (is_a<function>(e)) {
1686                         std::string name = ex_to<function>(e).get_name();
1687                         if (name == "H") {
1688
1689                                 lst parameter = ex_to<lst>(e.op(0));
1690                                 ex arg = e.op(1);
1691
1692                                 // special cases if all parameters are either 0, 1 or -1
1693                                 bool allthesame = true;
1694                                 if (parameter.op(0) == 0) {
1695                                         for (int i=1; i<parameter.nops(); i++) {
1696                                                 if (parameter.op(i) != 0) {
1697                                                         allthesame = false;
1698                                                         break;
1699                                                 }
1700                                         }
1701                                         if (allthesame) {
1702                                                 map_trafo_H_mult unify;
1703                                                 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1704                                                        / factorial(parameter.nops())).expand());
1705                                         }
1706                                 } else if (parameter.op(0) == -1) {
1707                                         for (int i=1; i<parameter.nops(); i++) {
1708                                                 if (parameter.op(i) != -1) {
1709                                                         allthesame = false;
1710                                                         break;
1711                                                 }
1712                                         }
1713                                         if (allthesame) {
1714                                                 map_trafo_H_mult unify;
1715                                                 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1716                                                        / factorial(parameter.nops())).expand());
1717                                         }
1718                                 } else {
1719                                         for (int i=1; i<parameter.nops(); i++) {
1720                                                 if (parameter.op(i) != 1) {
1721                                                         allthesame = false;
1722                                                         break;
1723                                                 }
1724                                         }
1725                                         if (allthesame) {
1726                                                 map_trafo_H_mult unify;
1727                                                 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1728                                                        / factorial(parameter.nops())).expand());
1729                                         }
1730                                 }
1731
1732                                 lst newparameter = parameter;
1733                                 newparameter.remove_first();
1734
1735                                 if (parameter.op(0) == 0) {
1736
1737                                         // leading zero
1738                                         ex res = convert_H_to_zeta(parameter);
1739                                         map_trafo_H_1mxt1px recursion;
1740                                         ex buffer = recursion(H(newparameter, arg).hold());
1741                                         if (is_a<add>(buffer)) {
1742                                                 for (int i=0; i<buffer.nops(); i++) {
1743                                                         res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1744                                                 }
1745                                         } else {
1746                                                 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1747                                         }
1748                                         return res;
1749
1750                                 } else if (parameter.op(0) == -1) {
1751
1752                                         // leading negative one
1753                                         ex res = convert_H_to_zeta(parameter);
1754                                         map_trafo_H_1mxt1px recursion;
1755                                         ex buffer = recursion(H(newparameter, arg).hold());
1756                                         if (is_a<add>(buffer)) {
1757                                                 for (int i=0; i<buffer.nops(); i++) {
1758                                                         res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1759                                                 }
1760                                         } else {
1761                                                 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1762                                         }
1763                                         return res;
1764
1765                                 } else {
1766
1767                                         // leading one
1768                                         map_trafo_H_1mxt1px recursion;
1769                                         map_trafo_H_mult unify;
1770                                         ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1771                                         int firstzero = 0;
1772                                         while (parameter.op(firstzero) == 1) {
1773                                                 firstzero++;
1774                                         }
1775                                         for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1776                                                 lst newparameter;
1777                                                 int j=0;
1778                                                 for (; j<=i; j++) {
1779                                                         newparameter.append(parameter[j+1]);
1780                                                 }
1781                                                 newparameter.append(1);
1782                                                 for (; j<parameter.nops()-1; j++) {
1783                                                         newparameter.append(parameter[j+1]);
1784                                                 }
1785                                                 res -= H(newparameter, arg).hold();
1786                                         }
1787                                         res = recursion(res).expand() / firstzero;
1788                                         return unify(res);
1789
1790                                 }
1791
1792                         }
1793                 }
1794                 return e;
1795         }
1796 };
1797
1798
1799 // do the actual summation.
1800 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1801 {
1802         const int j = m.size();
1803
1804         std::vector<cln::cl_N> t(j);
1805
1806         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1807         cln::cl_N factor = cln::expt(x, j) * one;
1808         cln::cl_N t0buf;
1809         int q = 0;
1810         do {
1811                 t0buf = t[0];
1812                 q++;
1813                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1814                 for (int k=j-2; k>=1; k--) {
1815                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1816                 }
1817                 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1818                 factor = factor * x;
1819         } while (t[0] != t0buf);
1820
1821         return t[0];
1822 }
1823
1824
1825 } // end of anonymous namespace
1826
1827
1828 //////////////////////////////////////////////////////////////////////
1829 //
1830 // Harmonic polylogarithm  H(m,x)
1831 //
1832 // GiNaC function
1833 //
1834 //////////////////////////////////////////////////////////////////////
1835
1836
1837 static ex H_evalf(const ex& x1, const ex& x2)
1838 {
1839         if (is_a<lst>(x1) && is_a<numeric>(x2)) {
1840                 for (int i=0; i<x1.nops(); i++) {
1841                         if (!x1.op(i).info(info_flags::integer)) {
1842                                 return H(x1,x2).hold();
1843                         }
1844                 }
1845                 if (x1.nops() < 1) {
1846                         return H(x1,x2).hold();
1847                 }
1848
1849                 cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
1850
1851                 const lst& morg = ex_to<lst>(x1);
1852                 // remove trailing zeros ...
1853                 if (*(--morg.end()) == 0) {
1854                         symbol xtemp("xtemp");
1855                         map_trafo_H_reduce_trailing_zeros filter;
1856                         return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
1857                 }
1858                 // ... and expand parameter notation
1859                 lst m;
1860                 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
1861                         if (*it > 1) {
1862                                 for (ex count=*it-1; count > 0; count--) {
1863                                         m.append(0);
1864                                 }
1865                                 m.append(1);
1866                         } else if (*it < -1) {
1867                                 for (ex count=*it+1; count < 0; count++) {
1868                                         m.append(0);
1869                                 }
1870                                 m.append(-1);
1871                         } else {
1872                                 m.append(*it);
1873                         }
1874                 }
1875
1876                 // since the transformations produce a lot of terms, they are only efficient for
1877                 // argument near one.
1878                 // no transformation needed -> do summation
1879                 if (cln::abs(x) < 0.95) {
1880                         lst m_lst;
1881                         lst s_lst;
1882                         ex pf;
1883                         if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
1884                                 // negative parameters -> s_lst is filled
1885                                 std::vector<int> m_int;
1886                                 std::vector<cln::cl_N> x_cln;
1887                                 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
1888                                      it_int != m_lst.end(); it_int++, it_cln++) {
1889                                         m_int.push_back(ex_to<numeric>(*it_int).to_int());
1890                                         x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
1891                                 }
1892                                 x_cln.front() = x_cln.front() * x;
1893                                 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
1894                         } else {
1895                                 // only positive parameters
1896                                 //TODO
1897                                 if (m_lst.nops() == 1) {
1898                                         return Li(m_lst.op(0), x2).evalf();
1899                                 }
1900                                 std::vector<int> m_int;
1901                                 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
1902                                         m_int.push_back(ex_to<numeric>(*it).to_int());
1903                                 }
1904                                 return numeric(H_do_sum(m_int, x));
1905                         }
1906                 }
1907
1908                 ex res = 1;
1909
1910                 // ensure that the realpart of the argument is positive
1911                 if (cln::realpart(x) < 0) {
1912                         x = -x;
1913                         for (int i=0; i<m.nops(); i++) {
1914                                 if (m.op(i) != 0) {
1915                                         m.let_op(i) = -m.op(i);
1916                                         res *= -1;
1917                                 }
1918                         }
1919                 }
1920
1921                 // choose transformations
1922                 symbol xtemp("xtemp");
1923                 if (cln::abs(x-1) < 1.4142) {
1924                         // x -> (1-x)/(1+x)
1925                         map_trafo_H_1mxt1px trafo;
1926                         res *= trafo(H(m, xtemp));
1927                 } else {
1928                         // x -> 1/x
1929                         map_trafo_H_1overx trafo;
1930                         res *= trafo(H(m, xtemp));
1931                 }
1932
1933                 // simplify result
1934 // TODO
1935 //              map_trafo_H_convert converter;
1936 //              res = converter(res).expand();
1937 //              lst ll;
1938 //              res.find(H(wild(1),wild(2)), ll);
1939 //              res.find(zeta(wild(1)), ll);
1940 //              res.find(zeta(wild(1),wild(2)), ll);
1941 //              res = res.collect(ll);
1942
1943                 return res.subs(xtemp == numeric(x)).evalf();
1944         }
1945
1946         return H(x1,x2).hold();
1947 }
1948
1949
1950 static ex H_eval(const ex& m_, const ex& x)
1951 {
1952         lst m;
1953         if (is_a<lst>(m_)) {
1954                 m = ex_to<lst>(m_);
1955         } else {
1956                 m = lst(m_);
1957         }
1958         if (m.nops() == 0) {
1959                 return _ex1;
1960         }
1961         ex pos1;
1962         ex pos2;
1963         ex n;
1964         ex p;
1965         int step = 0;
1966         if (*m.begin() > _ex1) {
1967                 step++;
1968                 pos1 = _ex0;
1969                 pos2 = _ex1;
1970                 n = *m.begin()-1;
1971                 p = _ex1;
1972         } else if (*m.begin() < _ex_1) {
1973                 step++;
1974                 pos1 = _ex0;
1975                 pos2 = _ex_1;
1976                 n = -*m.begin()-1;
1977                 p = _ex1;
1978         } else if (*m.begin() == _ex0) {
1979                 pos1 = _ex0;
1980                 n = _ex1;
1981         } else {
1982                 pos1 = *m.begin();
1983                 p = _ex1;
1984         }
1985         for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
1986                 if ((*it).info(info_flags::integer)) {
1987                         if (step == 0) {
1988                                 if (*it > _ex1) {
1989                                         if (pos1 == _ex0) {
1990                                                 step = 1;
1991                                                 pos2 = _ex1;
1992                                                 n += *it-1;
1993                                                 p = _ex1;
1994                                         } else {
1995                                                 step = 2;
1996                                         }
1997                                 } else if (*it < _ex_1) {
1998                                         if (pos1 == _ex0) {
1999                                                 step = 1;
2000                                                 pos2 = _ex_1;
2001                                                 n += -*it-1;
2002                                                 p = _ex1;
2003                                         } else {
2004                                                 step = 2;
2005                                         }
2006                                 } else {
2007                                         if (*it != pos1) {
2008                                                 step = 1;
2009                                                 pos2 = *it;
2010                                         }
2011                                         if (*it == _ex0) {
2012                                                 n++;
2013                                         } else {
2014                                                 p++;
2015                                         }
2016                                 }
2017                         } else if (step == 1) {
2018                                 if (*it != pos2) {
2019                                         step = 2;
2020                                 } else {
2021                                         if (*it == _ex0) {
2022                                                 n++;
2023                                         } else {
2024                                                 p++;
2025                                         }
2026                                 }
2027                         }
2028                 } else {
2029                         // if some m_i is not an integer
2030                         return H(m_, x).hold();
2031                 }
2032         }
2033         if ((x == _ex1) && (*(--m.end()) != _ex0)) {
2034                 return convert_H_to_zeta(m);
2035         }
2036         if (step == 0) {
2037                 if (pos1 == _ex0) {
2038                         // all zero
2039                         if (x == _ex0) {
2040                                 return H(m_, x).hold();
2041                         }
2042                         return pow(log(x), m.nops()) / factorial(m.nops());
2043                 } else {
2044                         // all (minus) one
2045                         return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
2046                 }
2047         } else if ((step == 1) && (pos1 == _ex0)){
2048                 // convertible to S
2049                 if (pos2 == _ex1) {
2050                         return S(n, p, x);
2051                 } else {
2052                         return pow(-1, p) * S(n, p, -x);
2053                 }
2054         }
2055         if (x == _ex0) {
2056                 return _ex0;
2057         }
2058         if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2059                 return H(m_, x).evalf();
2060         }
2061         return H(m_, x).hold();
2062 }
2063
2064
2065 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
2066 {
2067         epvector seq;
2068         seq.push_back(expair(H(m, x), 0));
2069         return pseries(rel, seq);
2070 }
2071
2072
2073 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
2074 {
2075         GINAC_ASSERT(deriv_param < 2);
2076         if (deriv_param == 0) {
2077                 return _ex0;
2078         }
2079         lst m;
2080         if (is_a<lst>(m_)) {
2081                 m = ex_to<lst>(m_);
2082         } else {
2083                 m = lst(m_);
2084         }
2085         ex mb = *m.begin();
2086         if (mb > _ex1) {
2087                 m[0]--;
2088                 return H(m, x) / x;
2089         }
2090         if (mb < _ex_1) {
2091                 m[0]++;
2092                 return H(m, x) / x;
2093         }
2094         m.remove_first();
2095         if (mb == _ex1) {
2096                 return 1/(1-x) * H(m, x);
2097         } else if (mb == _ex_1) {
2098                 return 1/(1+x) * H(m, x);
2099         } else {
2100                 return H(m, x) / x;
2101         }
2102 }
2103
2104
2105 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
2106 {
2107         lst m;
2108         if (is_a<lst>(m_)) {
2109                 m = ex_to<lst>(m_);
2110         } else {
2111                 m = lst(m_);
2112         }
2113         c.s << "\\mbox{H}_{";
2114         lst::const_iterator itm = m.begin();
2115         (*itm).print(c);
2116         itm++;
2117         for (; itm != m.end(); itm++) {
2118                 c.s << ",";
2119                 (*itm).print(c);
2120         }
2121         c.s << "}(";
2122         x.print(c);
2123         c.s << ")";
2124 }
2125
2126
2127 REGISTER_FUNCTION(H,
2128                   evalf_func(H_evalf).
2129                   eval_func(H_eval).
2130                   series_func(H_series).
2131                   derivative_func(H_deriv).
2132                   print_func<print_latex>(H_print_latex).
2133                   do_not_evalf_params());
2134
2135
2136 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
2137 ex convert_H_to_Li(const ex& m, const ex& x)
2138 {
2139         map_trafo_H_reduce_trailing_zeros filter;
2140         map_trafo_H_convert_to_Li filter2;
2141         if (is_a<lst>(m)) {
2142                 return filter2(filter(H(m, x).hold()));
2143         } else {
2144                 return filter2(filter(H(lst(m), x).hold()));
2145         }
2146 }
2147
2148
2149 //////////////////////////////////////////////////////////////////////
2150 //
2151 // Multiple zeta values  zeta(x) and zeta(x,s)
2152 //
2153 // helper functions
2154 //
2155 //////////////////////////////////////////////////////////////////////
2156
2157
2158 // anonymous namespace for helper functions
2159 namespace {
2160
2161
2162 // parameters and data for [Cra] algorithm
2163 const cln::cl_N lambda = cln::cl_N("319/320");
2164 int L1;
2165 int L2;
2166 std::vector<std::vector<cln::cl_N> > f_kj;
2167 std::vector<cln::cl_N> crB;
2168 std::vector<std::vector<cln::cl_N> > crG;
2169 std::vector<cln::cl_N> crX;
2170
2171
2172 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
2173 {
2174         const int size = a.size();
2175         for (int n=0; n<size; n++) {
2176                 c[n] = 0;
2177                 for (int m=0; m<=n; m++) {
2178                         c[n] = c[n] + a[m]*b[n-m];
2179                 }
2180         }
2181 }
2182
2183
2184 // [Cra] section 4
2185 void initcX(const std::vector<int>& s)
2186 {
2187         const int k = s.size();
2188
2189         crX.clear();
2190         crG.clear();
2191         crB.clear();
2192
2193         for (int i=0; i<=L2; i++) {
2194                 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
2195         }
2196
2197         int Sm = 0;
2198         int Smp1 = 0;
2199         for (int m=0; m<k-1; m++) {
2200                 std::vector<cln::cl_N> crGbuf;
2201                 Sm = Sm + s[m];
2202                 Smp1 = Sm + s[m+1];
2203                 for (int i=0; i<=L2; i++) {
2204                         crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
2205                 }
2206                 crG.push_back(crGbuf);
2207         }
2208
2209         crX = crB;
2210
2211         for (int m=0; m<k-1; m++) {
2212                 std::vector<cln::cl_N> Xbuf;
2213                 for (int i=0; i<=L2; i++) {
2214                         Xbuf.push_back(crX[i] * crG[m][i]);
2215                 }
2216                 halfcyclic_convolute(Xbuf, crB, crX);
2217         }
2218 }
2219
2220
2221 // [Cra] section 4
2222 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
2223 {
2224         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2225         cln::cl_N factor = cln::expt(lambda, Sqk);
2226         cln::cl_N res = factor / Sqk * crX[0] * one;
2227         cln::cl_N resbuf;
2228         int N = 0;
2229         do {
2230                 resbuf = res;
2231                 factor = factor * lambda;
2232                 N++;
2233                 res = res + crX[N] * factor / (N+Sqk);
2234         } while ((res != resbuf) || cln::zerop(crX[N]));
2235         return res;
2236 }
2237
2238
2239 // [Cra] section 4
2240 void calc_f(int maxr)
2241 {
2242         f_kj.clear();
2243         f_kj.resize(L1);
2244
2245         cln::cl_N t0, t1, t2, t3, t4;
2246         int i, j, k;
2247         std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
2248         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2249
2250         t0 = cln::exp(-lambda);
2251         t2 = 1;
2252         for (k=1; k<=L1; k++) {
2253                 t1 = k * lambda;
2254                 t2 = t0 * t2;
2255                 for (j=1; j<=maxr; j++) {
2256                         t3 = 1;
2257                         t4 = 1;
2258                         for (i=2; i<=j; i++) {
2259                                 t4 = t4 * (j-i+1);
2260                                 t3 = t1 * t3 + t4;
2261                         }
2262                         (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
2263                 }
2264                 it++;
2265         }
2266 }
2267
2268
2269 // [Cra] (3.1)
2270 cln::cl_N crandall_Z(const std::vector<int>& s)
2271 {
2272         const int j = s.size();
2273
2274         if (j == 1) {
2275                 cln::cl_N t0;
2276                 cln::cl_N t0buf;
2277                 int q = 0;
2278                 do {
2279                         t0buf = t0;
2280                         q++;
2281                         t0 = t0 + f_kj[q+j-2][s[0]-1];
2282                 } while (t0 != t0buf);
2283
2284                 return t0 / cln::factorial(s[0]-1);
2285         }
2286
2287         std::vector<cln::cl_N> t(j);
2288
2289         cln::cl_N t0buf;
2290         int q = 0;
2291         do {
2292                 t0buf = t[0];
2293                 q++;
2294                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
2295                 for (int k=j-2; k>=1; k--) {
2296                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
2297                 }
2298                 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
2299         } while (t[0] != t0buf);
2300
2301         return t[0] / cln::factorial(s[0]-1);
2302 }
2303
2304
2305 // [Cra] (2.4)
2306 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
2307 {
2308         std::vector<int> r = s;
2309         const int j = r.size();
2310
2311         // decide on maximal size of f_kj for crandall_Z
2312         if (Digits < 50) {
2313                 L1 = 150;
2314         } else {
2315                 L1 = Digits * 3 + j*2;
2316         }
2317
2318         // decide on maximal size of crX for crandall_Y
2319         if (Digits < 38) {
2320                 L2 = 63;
2321         } else if (Digits < 86) {
2322                 L2 = 127;
2323         } else if (Digits < 192) {
2324                 L2 = 255;
2325         } else if (Digits < 394) {
2326                 L2 = 511;
2327         } else if (Digits < 808) {
2328                 L2 = 1023;
2329         } else {
2330                 L2 = 2047;
2331         }
2332
2333         cln::cl_N res;
2334
2335         int maxr = 0;
2336         int S = 0;
2337         for (int i=0; i<j; i++) {
2338                 S += r[i];
2339                 if (r[i] > maxr) {
2340                         maxr = r[i];
2341                 }
2342         }
2343
2344         calc_f(maxr);
2345
2346         const cln::cl_N r0factorial = cln::factorial(r[0]-1);
2347
2348         std::vector<int> rz;
2349         int skp1buf;
2350         int Srun = S;
2351         for (int k=r.size()-1; k>0; k--) {
2352
2353                 rz.insert(rz.begin(), r.back());
2354                 skp1buf = rz.front();
2355                 Srun -= skp1buf;
2356                 r.pop_back();
2357
2358                 initcX(r);
2359
2360                 for (int q=0; q<skp1buf; q++) {
2361
2362                         cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2363                         cln::cl_N pp2 = crandall_Z(rz);
2364
2365                         rz.front()--;
2366
2367                         if (q & 1) {
2368                                 res = res - pp1 * pp2 / cln::factorial(q);
2369                         } else {
2370                                 res = res + pp1 * pp2 / cln::factorial(q);
2371                         }
2372                 }
2373                 rz.front() = skp1buf;
2374         }
2375         rz.insert(rz.begin(), r.back());
2376
2377         initcX(rz);
2378
2379         res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2380
2381         return res;
2382 }
2383
2384
2385 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2386 {
2387         const int j = r.size();
2388
2389         // buffer for subsums
2390         std::vector<cln::cl_N> t(j);
2391         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2392
2393         cln::cl_N t0buf;
2394         int q = 0;
2395         do {
2396                 t0buf = t[0];
2397                 q++;
2398                 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2399                 for (int k=j-2; k>=0; k--) {
2400                         t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2401                 }
2402         } while (t[0] != t0buf);
2403
2404         return t[0];
2405 }
2406
2407
2408 // does Hoelder convolution. see [BBB] (7.0)
2409 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2410 {
2411         // prepare parameters
2412         // holds Li arguments in [BBB] notation
2413         std::vector<int> s = s_;
2414         std::vector<int> m_p = m_;
2415         std::vector<int> m_q;
2416         // holds Li arguments in nested sums notation
2417         std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2418         s_p[0] = s_p[0] * cln::cl_N("1/2");
2419         // convert notations
2420         int sig = 1;
2421         for (int i=0; i<s_.size(); i++) {
2422                 if (s_[i] < 0) {
2423                         sig = -sig;
2424                         s_p[i] = -s_p[i];
2425                 }
2426                 s[i] = sig * std::abs(s[i]);
2427         }
2428         std::vector<cln::cl_N> s_q;
2429         cln::cl_N signum = 1;
2430
2431         // first term
2432         cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2433
2434         // middle terms
2435         do {
2436
2437                 // change parameters
2438                 if (s.front() > 0) {
2439                         if (m_p.front() == 1) {
2440                                 m_p.erase(m_p.begin());
2441                                 s_p.erase(s_p.begin());
2442                                 if (s_p.size() > 0) {
2443                                         s_p.front() = s_p.front() * cln::cl_N("1/2");
2444                                 }
2445                                 s.erase(s.begin());
2446                                 m_q.front()++;
2447                         } else {
2448                                 m_p.front()--;
2449                                 m_q.insert(m_q.begin(), 1);
2450                                 if (s_q.size() > 0) {
2451                                         s_q.front() = s_q.front() * 2;
2452                                 }
2453                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2454                         }
2455                 } else {
2456                         if (m_p.front() == 1) {
2457                                 m_p.erase(m_p.begin());
2458                                 cln::cl_N spbuf = s_p.front();
2459                                 s_p.erase(s_p.begin());
2460                                 if (s_p.size() > 0) {
2461                                         s_p.front() = s_p.front() * spbuf;
2462                                 }
2463                                 s.erase(s.begin());
2464                                 m_q.insert(m_q.begin(), 1);
2465                                 if (s_q.size() > 0) {
2466                                         s_q.front() = s_q.front() * 4;
2467                                 }
2468                                 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2469                                 signum = -signum;
2470                         } else {
2471                                 m_p.front()--;
2472                                 m_q.insert(m_q.begin(), 1);
2473                                 if (s_q.size() > 0) {
2474                                         s_q.front() = s_q.front() * 2;
2475                                 }
2476                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2477                         }
2478                 }
2479
2480                 // exiting the loop
2481                 if (m_p.size() == 0) break;
2482
2483                 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2484
2485         } while (true);
2486
2487         // last term
2488         res = res + signum * multipleLi_do_sum(m_q, s_q);
2489
2490         return res;
2491 }
2492
2493
2494 } // end of anonymous namespace
2495
2496
2497 //////////////////////////////////////////////////////////////////////
2498 //
2499 // Multiple zeta values  zeta(x)
2500 //
2501 // GiNaC function
2502 //
2503 //////////////////////////////////////////////////////////////////////
2504
2505
2506 static ex zeta1_evalf(const ex& x)
2507 {
2508         if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2509
2510                 // multiple zeta value
2511                 const int count = x.nops();
2512                 const lst& xlst = ex_to<lst>(x);
2513                 std::vector<int> r(count);
2514
2515                 // check parameters and convert them
2516                 lst::const_iterator it1 = xlst.begin();
2517                 std::vector<int>::iterator it2 = r.begin();
2518                 do {
2519                         if (!(*it1).info(info_flags::posint)) {
2520                                 return zeta(x).hold();
2521                         }
2522                         *it2 = ex_to<numeric>(*it1).to_int();
2523                         it1++;
2524                         it2++;
2525                 } while (it2 != r.end());
2526
2527                 // check for divergence
2528                 if (r[0] == 1) {
2529                         return zeta(x).hold();
2530                 }
2531
2532                 // decide on summation algorithm
2533                 // this is still a bit clumsy
2534                 int limit = (Digits>17) ? 10 : 6;
2535                 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2536                         return numeric(zeta_do_sum_Crandall(r));
2537                 } else {
2538                         return numeric(zeta_do_sum_simple(r));
2539                 }
2540         }
2541
2542         // single zeta value
2543         if (is_exactly_a<numeric>(x) && (x != 1)) {
2544                 try {
2545                         return zeta(ex_to<numeric>(x));
2546                 } catch (const dunno &e) { }
2547         }
2548
2549         return zeta(x).hold();
2550 }
2551
2552
2553 static ex zeta1_eval(const ex& m)
2554 {
2555         if (is_exactly_a<lst>(m)) {
2556                 if (m.nops() == 1) {
2557                         return zeta(m.op(0));
2558                 }
2559                 return zeta(m).hold();
2560         }
2561
2562         if (m.info(info_flags::numeric)) {
2563                 const numeric& y = ex_to<numeric>(m);
2564                 // trap integer arguments:
2565                 if (y.is_integer()) {
2566                         if (y.is_zero()) {
2567                                 return _ex_1_2;
2568                         }
2569                         if (y.is_equal(_num1)) {
2570                                 return zeta(m).hold();
2571                         }
2572                         if (y.info(info_flags::posint)) {
2573                                 if (y.info(info_flags::odd)) {
2574                                         return zeta(m).hold();
2575                                 } else {
2576                                         return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2577                                 }
2578                         } else {
2579                                 if (y.info(info_flags::odd)) {
2580                                         return -bernoulli(_num1-y) / (_num1-y);
2581                                 } else {
2582                                         return _ex0;
2583                                 }
2584                         }
2585                 }
2586                 // zeta(float)
2587                 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
2588                         return zeta1_evalf(m);
2589                 }
2590         }
2591         return zeta(m).hold();
2592 }
2593
2594
2595 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
2596 {
2597         GINAC_ASSERT(deriv_param==0);
2598
2599         if (is_exactly_a<lst>(m)) {
2600                 return _ex0;
2601         } else {
2602                 return zetaderiv(_ex1, m);
2603         }
2604 }
2605
2606
2607 static void zeta1_print_latex(const ex& m_, const print_context& c)
2608 {
2609         c.s << "\\zeta(";
2610         if (is_a<lst>(m_)) {
2611                 const lst& m = ex_to<lst>(m_);
2612                 lst::const_iterator it = m.begin();
2613                 (*it).print(c);
2614                 it++;
2615                 for (; it != m.end(); it++) {
2616                         c.s << ",";
2617                         (*it).print(c);
2618                 }
2619         } else {
2620                 m_.print(c);
2621         }
2622         c.s << ")";
2623 }
2624
2625
2626 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta").
2627                                 evalf_func(zeta1_evalf).
2628                                 eval_func(zeta1_eval).
2629                                 derivative_func(zeta1_deriv).
2630                                 print_func<print_latex>(zeta1_print_latex).
2631                                 do_not_evalf_params().
2632                                 overloaded(2));
2633
2634
2635 //////////////////////////////////////////////////////////////////////
2636 //
2637 // Alternating Euler sum  zeta(x,s)
2638 //
2639 // GiNaC function
2640 //
2641 //////////////////////////////////////////////////////////////////////
2642
2643
2644 static ex zeta2_evalf(const ex& x, const ex& s)
2645 {
2646         if (is_exactly_a<lst>(x)) {
2647
2648                 // alternating Euler sum
2649                 const int count = x.nops();
2650                 const lst& xlst = ex_to<lst>(x);
2651                 const lst& slst = ex_to<lst>(s);
2652                 std::vector<int> xi(count);
2653                 std::vector<int> si(count);
2654
2655                 // check parameters and convert them
2656                 lst::const_iterator it_xread = xlst.begin();
2657                 lst::const_iterator it_sread = slst.begin();
2658                 std::vector<int>::iterator it_xwrite = xi.begin();
2659                 std::vector<int>::iterator it_swrite = si.begin();
2660                 do {
2661                         if (!(*it_xread).info(info_flags::posint)) {
2662                                 return zeta(x, s).hold();
2663                         }
2664                         *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2665                         if (*it_sread > 0) {
2666                                 *it_swrite = 1;
2667                         } else {
2668                                 *it_swrite = -1;
2669                         }
2670                         it_xread++;
2671                         it_sread++;
2672                         it_xwrite++;
2673                         it_swrite++;
2674                 } while (it_xwrite != xi.end());
2675
2676                 // check for divergence
2677                 if ((xi[0] == 1) && (si[0] == 1)) {
2678                         return zeta(x, s).hold();
2679                 }
2680
2681                 // use Hoelder convolution
2682                 return numeric(zeta_do_Hoelder_convolution(xi, si));
2683         }
2684
2685         return zeta(x, s).hold();
2686 }
2687
2688
2689 static ex zeta2_eval(const ex& m, const ex& s_)
2690 {
2691         if (is_exactly_a<lst>(s_)) {
2692                 const lst& s = ex_to<lst>(s_);
2693                 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
2694                         if ((*it).info(info_flags::positive)) {
2695                                 continue;
2696                         }
2697                         return zeta(m, s_).hold();
2698                 }
2699                 return zeta(m);
2700         } else if (s_.info(info_flags::positive)) {
2701                 return zeta(m);
2702         }
2703
2704         return zeta(m, s_).hold();
2705 }
2706
2707
2708 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
2709 {
2710         GINAC_ASSERT(deriv_param==0);
2711
2712         if (is_exactly_a<lst>(m)) {
2713                 return _ex0;
2714         } else {
2715                 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
2716                         return zetaderiv(_ex1, m);
2717                 }
2718                 return _ex0;
2719         }
2720 }
2721
2722
2723 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
2724 {
2725         lst m;
2726         if (is_a<lst>(m_)) {
2727                 m = ex_to<lst>(m_);
2728         } else {
2729                 m = lst(m_);
2730         }
2731         lst s;
2732         if (is_a<lst>(s_)) {
2733                 s = ex_to<lst>(s_);
2734         } else {
2735                 s = lst(s_);
2736         }
2737         c.s << "\\zeta(";
2738         lst::const_iterator itm = m.begin();
2739         lst::const_iterator its = s.begin();
2740         if (*its < 0) {
2741                 c.s << "\\overline{";
2742                 (*itm).print(c);
2743                 c.s << "}";
2744         } else {
2745                 (*itm).print(c);
2746         }
2747         its++;
2748         itm++;
2749         for (; itm != m.end(); itm++, its++) {
2750                 c.s << ",";
2751                 if (*its < 0) {
2752                         c.s << "\\overline{";
2753                         (*itm).print(c);
2754                         c.s << "}";
2755                 } else {
2756                         (*itm).print(c);
2757                 }
2758         }
2759         c.s << ")";
2760 }
2761
2762
2763 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta").
2764                                 evalf_func(zeta2_evalf).
2765                                 eval_func(zeta2_eval).
2766                                 derivative_func(zeta2_deriv).
2767                                 print_func<print_latex>(zeta2_print_latex).
2768                                 do_not_evalf_params().
2769                                 overloaded(2));
2770
2771
2772 } // namespace GiNaC
2773