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 speeded 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) 1999-2004 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 // initial size of Xn that should suffice for 32bit machines (must be even)
 106 const int xninitsizestep = 26;
 107 int xninitsize = xninitsizestep;
 108 int xnsize = 0;
 109
 110
 111 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
 112 // With these numbers the polylogs can be calculated as follows:
 113 //   Li_p (x)  =  \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with  u = -log(1-x)
 114 //   X_0(n) = B_n (Bernoulli numbers)
 115 //   X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
 116 // The calculation of Xn depends on X0 and X{n-1}.
 117 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
 118 // This results in a slightly more complicated algorithm for the X_n.
 119 // The first index in Xn corresponds to the index of the polylog minus 2.
 120 // The second index in Xn corresponds to the index from the actual sum.
 121 void fill_Xn(int n)
 122 {
 123         if (n>1) {
 124                 // calculate X_2 and higher (corresponding to Li_4 and higher)
 125                 std::vector<cln::cl_N> buf(xninitsize);
 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<=xninitsize; 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(xninitsize);
 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<=xninitsize; 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(xninitsize/2);
 175                 std::vector<cln::cl_N>::iterator it = buf.begin();
 176                 for (int i=1; i<=xninitsize/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 // doubles the number of entries in each Xn[]
 188 void double_Xn()
 189 {
 190         const int pos0 = xninitsize / 2;
 191         // X_0
 192         for (int i=1; i<=xninitsizestep/2; ++i) {
 193                 Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
 194         }
 195         if (Xn.size() > 0) {
 196                 int xend = xninitsize + xninitsizestep;
 197                 cln::cl_N result;
 198                 // X_1
 199                 for (int i=xninitsize+1; i<=xend; ++i) {
 200                         if (i & 1) {
 201                                 result = -Xn[0][(i-3)/2]/2;
 202                                 Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result);
 203                         } else {
 204                                 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
 205                                 for (int k=1; k<i/2; k++) {
 206                                         result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
 207                                 }
 208                                 Xn[1].push_back(result);
 209                         }
 210                 }
 211                 // X_n
 212                 for (int n=2; n<Xn.size(); ++n) {
 213                         for (int i=xninitsize+1; i<=xend; ++i) {
 214                                 if (i & 1) {
 215                                         result = 0; // k == 0
 216                                 } else {
 217                                         result = Xn[0][i/2-1]; // k == 0
 218                                 }
 219                                 for (int k=1; k<i-1; ++k) {
 220                                         if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
 221                                                 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
 222                                         }
 223                                 }
 224                                 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
 225                                 result = result + Xn[n-1][i-1] / (i+1); // k == i
 226                                 Xn[n].push_back(result);
 227                         }
 228                 }
 229         }
 230         xninitsize += xninitsizestep;
 231 }
 232
 233
 234 // calculates Li(2,x) without Xn
 235 cln::cl_N Li2_do_sum(const cln::cl_N& x)
 236 {
 237         cln::cl_N res = x;
 238         cln::cl_N resbuf;
 239         cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
 240         cln::cl_I den = 1; // n^2 = 1
 241         unsigned i = 3;
 242         do {
 243                 resbuf = res;
 244                 num = num * x;
 245                 den = den + i;  // n^2 = 4, 9, 16, ...
 246                 i += 2;
 247                 res = res + num / den;
 248         } while (res != resbuf);
 249         return res;
 250 }
 251
 252
 253 // calculates Li(2,x) with Xn
 254 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
 255 {
 256         std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
 257         std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
 258         cln::cl_N u = -cln::log(1-x);
 259         cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
 260         cln::cl_N res = u - u*u/4;
 261         cln::cl_N resbuf;
 262         unsigned i = 1;
 263         do {
 264                 resbuf = res;
 265                 factor = factor * u*u / (2*i * (2*i+1));
 266                 res = res + (*it) * factor;
 267                 i++;
 268                 if (++it == xend) {
 269                         double_Xn();
 270                         it = Xn[0].begin() + (i-1);
 271                         xend = Xn[0].end();
 272                 }
 273         } while (res != resbuf);
 274         return res;
 275 }
 276
 277
 278 // calculates Li(n,x), n>2 without Xn
 279 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
 280 {
 281         cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
 282         cln::cl_N res = x;
 283         cln::cl_N resbuf;
 284         int i=2;
 285         do {
 286                 resbuf = res;
 287                 factor = factor * x;
 288                 res = res + factor / cln::expt(cln::cl_I(i),n);
 289                 i++;
 290         } while (res != resbuf);
 291         return res;
 292 }
 293
 294
 295 // calculates Li(n,x), n>2 with Xn
 296 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
 297 {
 298         std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
 299         std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
 300         cln::cl_N u = -cln::log(1-x);
 301         cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
 302         cln::cl_N res = u;
 303         cln::cl_N resbuf;
 304         unsigned i=2;
 305         do {
 306                 resbuf = res;
 307                 factor = factor * u / i;
 308                 res = res + (*it) * factor;
 309                 i++;
 310                 if (++it == xend) {
 311                         double_Xn();
 312                         it = Xn[n-2].begin() + (i-2);
 313                         xend = Xn[n-2].end();
 314                 }
 315         } while (res != resbuf);
 316         return res;
 317 }
 318
 319
 320 // forward declaration needed by function Li_projection and C below
 321 numeric S_num(int n, int p, const numeric& x);
 322
 323
 324 // helper function for classical polylog Li
 325 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
 326 {
 327         // treat n=2 as special case
 328         if (n == 2) {
 329                 // check if precalculated X0 exists
 330                 if (xnsize == 0) {
 331                         fill_Xn(0);
 332                 }
 333
 334                 if (cln::realpart(x) < 0.5) {
 335                         // choose the faster algorithm
 336                         // the switching point was empirically determined. the optimal point
 337                         // depends on hardware, Digits, ... so an approx value is okay.
 338                         // it solves also the problem with precision due to the u=-log(1-x) transformation
 339                         if (cln::abs(cln::realpart(x)) < 0.25) {
 340
 341                                 return Li2_do_sum(x);
 342                         } else {
 343                                 return Li2_do_sum_Xn(x);
 344                         }
 345                 } else {
 346                         // choose the faster algorithm
 347                         if (cln::abs(cln::realpart(x)) > 0.75) {
 348                                 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
 349                         } else {
 350                                 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
 351                         }
 352                 }
 353         } else {
 354                 // check if precalculated Xn exist
 355                 if (n > xnsize+1) {
 356                         for (int i=xnsize; i<n-1; i++) {
 357                                 fill_Xn(i);
 358                         }
 359                 }
 360
 361                 if (cln::realpart(x) < 0.5) {
 362                         // choose the faster algorithm
 363                         // with n>=12 the "normal" summation always wins against the method with Xn
 364                         if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
 365                                 return Lin_do_sum(n, x);
 366                         } else {
 367                                 return Lin_do_sum_Xn(n, x);
 368                         }
 369                 } else {
 370                         cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
 371                         for (int j=0; j<n-1; j++) {
 372                                 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
 373                                                   * cln::expt(cln::log(x), j) / cln::factorial(j);
 374                         }
 375                         return result;
 376                 }
 377         }
 378 }
 379
 380
 381 // helper function for classical polylog Li
 382 numeric Li_num(int n, const numeric& x)
 383 {
 384         if (n == 1) {
 385                 // just a log
 386                 return -cln::log(1-x.to_cl_N());
 387         }
 388         if (x.is_zero()) {
 389                 return 0;
 390         }
 391         if (x == 1) {
 392                 // [Kol] (2.22)
 393                 return cln::zeta(n);
 394         }
 395         else if (x == -1) {
 396                 // [Kol] (2.22)
 397                 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
 398         }
 399
 400         // what is the desired float format?
 401         // first guess: default format
 402         cln::float_format_t prec = cln::default_float_format;
 403         const cln::cl_N value = x.to_cl_N();
 404         // second guess: the argument's format
 405         if (!x.real().is_rational())
 406                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
 407         else if (!x.imag().is_rational())
 408                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
 409
 410         // [Kol] (5.15)
 411         if (cln::abs(value) > 1) {
 412                 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
 413                 // check if argument is complex. if it is real, the new polylog has to be conjugated.
 414                 if (cln::zerop(cln::imagpart(value))) {
 415                         if (n & 1) {
 416                                 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
 417                         }
 418                         else {
 419                                 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
 420                         }
 421                 }
 422                 else {
 423                         if (n & 1) {
 424                                 result = result + Li_projection(n, cln::recip(value), prec);
 425                         }
 426                         else {
 427                                 result = result - Li_projection(n, cln::recip(value), prec);
 428                         }
 429                 }
 430                 cln::cl_N add;
 431                 for (int j=0; j<n-1; j++) {
 432                         add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
 433                                     * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
 434                 }
 435                 result = result - add;
 436                 return result;
 437         }
 438         else {
 439                 return Li_projection(n, value, prec);
 440         }
 441 }
 442
 443
 444 } // end of anonymous namespace
 445
 446
 447 //////////////////////////////////////////////////////////////////////
 448 //
 449 // Multiple polylogarithm  Li(n,x)
 450 //
 451 // helper function
 452 //
 453 //////////////////////////////////////////////////////////////////////
 454
 455
 456 // anonymous namespace for helper function
 457 namespace {
 458
 459
 460 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
 461 {
 462         const int j = s.size();
 463
 464         std::vector<cln::cl_N> t(j);
 465         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
 466
 467         cln::cl_N t0buf;
 468         int q = 0;
 469         do {
 470                 t0buf = t[0];
 471                 // do it once ...
 472                 q++;
 473                 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
 474                 for (int k=j-2; k>=0; k--) {
 475                         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]);
 476                 }
 477                 // ... and do it again (to avoid premature drop out due to special arguments)
 478                 q++;
 479                 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
 480                 for (int k=j-2; k>=0; k--) {
 481                         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]);
 482                 }
 483         } while (t[0] != t0buf);
 484
 485         return t[0];
 486 }
 487
 488 // forward declaration for Li_eval()
 489 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
 490
 491
 492 } // end of anonymous namespace
 493
 494
 495 //////////////////////////////////////////////////////////////////////
 496 //
 497 // Classical polylogarithm and multiple polylogarithm  Li(n,x)
 498 //
 499 // GiNaC function
 500 //
 501 //////////////////////////////////////////////////////////////////////
 502
 503
 504 static ex Li_evalf(const ex& x1, const ex& x2)
 505 {
 506         // classical polylogs
 507         if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
 508                 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
 509         }
 510         // multiple polylogs
 511         else if (is_a<lst>(x1) && is_a<lst>(x2)) {
 512                 ex conv = 1;
 513                 for (int i=0; i<x1.nops(); i++) {
 514                         if (!x1.op(i).info(info_flags::posint)) {
 515                                 return Li(x1, x2).hold();
 516                         }
 517                         if (!is_a<numeric>(x2.op(i))) {
 518                                 return Li(x1, x2).hold();
 519                         }
 520                         conv *= x2.op(i);
 521                         if (abs(conv) >= 1) {
 522                                 return Li(x1, x2).hold();
 523                         }
 524                 }
 525
 526                 std::vector<int> m;
 527                 std::vector<cln::cl_N> x;
 528                 for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
 529                         m.push_back(ex_to<numeric>(x1.op(i)).to_int());
 530                         x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
 531                 }
 532
 533                 return numeric(multipleLi_do_sum(m, x));
 534         }
 535
 536         return Li(x1,x2).hold();
 537 }
 538
 539
 540 static ex Li_eval(const ex& m_, const ex& x_)
 541 {
 542         if (m_.nops() < 2) {
 543                 ex m;
 544                 if (is_a<lst>(m_)) {
 545                         m = m_.op(0);
 546                 } else {
 547                         m = m_;
 548                 }
 549                 ex x;
 550                 if (is_a<lst>(x_)) {
 551                         x = x_.op(0);
 552                 } else {
 553                         x = x_;
 554                 }
 555                 if (x == _ex0) {
 556                         return _ex0;
 557                 }
 558                 if (x == _ex1) {
 559                         return zeta(m);
 560                 }
 561                 if (x == _ex_1) {
 562                         return (pow(2,1-m)-1) * zeta(m);
 563                 }
 564                 if (m == _ex1) {
 565                         return -log(1-x);
 566                 }
 567                 if (m.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
 568                         return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
 569                 }
 570         } else {
 571                 bool ish = true;
 572                 bool iszeta = true;
 573                 bool iszero = false;
 574                 bool doevalf = false;
 575                 bool doevalfveto = true;
 576                 const lst& m = ex_to<lst>(m_);
 577                 const lst& x = ex_to<lst>(x_);
 578                 lst::const_iterator itm = m.begin();
 579                 lst::const_iterator itx = x.begin();
 580                 for (; itm != m.end(); itm++, itx++) {
 581                         if (!(*itm).info(info_flags::posint)) {
 582                                 return Li(m_, x_).hold();
 583                         }
 584                         if ((*itx != _ex1) && (*itx != _ex_1)) {
 585                                 if (itx != x.begin()) {
 586                                         ish = false;
 587                                 }
 588                                 iszeta = false;
 589                         }
 590                         if (*itx == _ex0) {
 591                                 iszero = true;
 592                         }
 593                         if (!(*itx).info(info_flags::numeric)) {
 594                                 doevalfveto = false;
 595                         }
 596                         if (!(*itx).info(info_flags::crational)) {
 597                                 doevalf = true;
 598                         }
 599                 }
 600                 if (iszeta) {
 601                         return zeta(m_, x_);
 602                 }
 603                 if (iszero) {
 604                         return _ex0;
 605                 }
 606                 if (ish) {
 607                         ex pf;
 608                         lst newm = convert_parameter_Li_to_H(m, x, pf);
 609                         return pf * H(newm, x[0]);
 610                 }
 611                 if (doevalfveto && doevalf) {
 612                         return Li(m_, x_).evalf();
 613                 }
 614         }
 615         return Li(m_, x_).hold();
 616 }
 617
 618
 619 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
 620 {
 621         epvector seq;
 622         seq.push_back(expair(Li(m, x), 0));
 623         return pseries(rel, seq);
 624 }
 625
 626
 627 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
 628 {
 629         GINAC_ASSERT(deriv_param < 2);
 630         if (deriv_param == 0) {
 631                 return _ex0;
 632         }
 633         if (m_.nops() > 1) {
 634                 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
 635         }
 636         ex m;
 637         if (is_a<lst>(m_)) {
 638                 m = m_.op(0);
 639         } else {
 640                 m = m_;
 641         }
 642         ex x;
 643         if (is_a<lst>(x_)) {
 644                 x = x_.op(0);
 645         } else {
 646                 x = x_;
 647         }
 648         if (m > 0) {
 649                 return Li(m-1, x) / x;
 650         } else {
 651                 return 1/(1-x);
 652         }
 653 }
 654
 655
 656 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
 657 {
 658         lst m;
 659         if (is_a<lst>(m_)) {
 660                 m = ex_to<lst>(m_);
 661         } else {
 662                 m = lst(m_);
 663         }
 664         lst x;
 665         if (is_a<lst>(x_)) {
 666                 x = ex_to<lst>(x_);
 667         } else {
 668                 x = lst(x_);
 669         }
 670         c.s << "\\mbox{Li}_{";
 671         lst::const_iterator itm = m.begin();
 672         (*itm).print(c);
 673         itm++;
 674         for (; itm != m.end(); itm++) {
 675                 c.s << ",";
 676                 (*itm).print(c);
 677         }
 678         c.s << "}(";
 679         lst::const_iterator itx = x.begin();
 680         (*itx).print(c);
 681         itx++;
 682         for (; itx != x.end(); itx++) {
 683                 c.s << ",";
 684                 (*itx).print(c);
 685         }
 686         c.s << ")";
 687 }
 688
 689
 690 REGISTER_FUNCTION(Li,
 691                   evalf_func(Li_evalf).
 692                   eval_func(Li_eval).
 693                   series_func(Li_series).
 694                   derivative_func(Li_deriv).
 695                   print_func<print_latex>(Li_print_latex).
 696                   do_not_evalf_params());
 697
 698
 699 //////////////////////////////////////////////////////////////////////
 700 //
 701 // Nielsen's generalized polylogarithm  S(n,p,x)
 702 //
 703 // helper functions
 704 //
 705 //////////////////////////////////////////////////////////////////////
 706
 707
 708 // anonymous namespace for helper functions
 709 namespace {
 710
 711
 712 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
 713 // see fill_Yn()
 714 std::vector<std::vector<cln::cl_N> > Yn;
 715 int ynsize = 0; // number of Yn[]
 716 int ynlength = 100; // initial length of all Yn[i]
 717
 718
 719 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
 720 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
 721 // representing S_{n,p}(x).
 722 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
 723 // equivalent Z-sum.
 724 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
 725 // representing S_{n,p}(x).
 726 // The calculation of Y_n uses the values from Y_{n-1}.
 727 void fill_Yn(int n, const cln::float_format_t& prec)
 728 {
 729         const int initsize = ynlength;
 730         //const int initsize = initsize_Yn;
 731         cln::cl_N one = cln::cl_float(1, prec);
 732
 733         if (n) {
 734                 std::vector<cln::cl_N> buf(initsize);
 735                 std::vector<cln::cl_N>::iterator it = buf.begin();
 736                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
 737                 *it = (*itprev) / cln::cl_N(n+1) * one;
 738                 it++;
 739                 itprev++;
 740                 // sums with an index smaller than the depth are zero and need not to be calculated.
 741                 // calculation starts with depth, which is n+2)
 742                 for (int i=n+2; i<=initsize+n; i++) {
 743                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
 744                         it++;
 745                         itprev++;
 746                 }
 747                 Yn.push_back(buf);
 748         } else {
 749                 std::vector<cln::cl_N> buf(initsize);
 750                 std::vector<cln::cl_N>::iterator it = buf.begin();
 751                 *it = 1 * one;
 752                 it++;
 753                 for (int i=2; i<=initsize; i++) {
 754                         *it = *(it-1) + 1 / cln::cl_N(i) * one;
 755                         it++;
 756                 }
 757                 Yn.push_back(buf);
 758         }
 759         ynsize++;
 760 }
 761
 762
 763 // make Yn longer ...
 764 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
 765 {
 766
 767         cln::cl_N one = cln::cl_float(1, prec);
 768
 769         Yn[0].resize(newsize);
 770         std::vector<cln::cl_N>::iterator it = Yn[0].begin();
 771         it += ynlength;
 772         for (int i=ynlength+1; i<=newsize; i++) {
 773                 *it = *(it-1) + 1 / cln::cl_N(i) * one;
 774                 it++;
 775         }
 776
 777         for (int n=1; n<ynsize; n++) {
 778                 Yn[n].resize(newsize);
 779                 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
 780                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
 781                 it += ynlength;
 782                 itprev += ynlength;
 783                 for (int i=ynlength+n+1; i<=newsize+n; i++) {
 784                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
 785                         it++;
 786                         itprev++;
 787                 }
 788         }
 789
 790         ynlength = newsize;
 791 }
 792
 793
 794 // helper function for S(n,p,x)
 795 // [Kol] (7.2)
 796 cln::cl_N C(int n, int p)
 797 {
 798         cln::cl_N result;
 799
 800         for (int k=0; k<p; k++) {
 801                 for (int j=0; j<=(n+k-1)/2; j++) {
 802                         if (k == 0) {
 803                                 if (n & 1) {
 804                                         if (j & 1) {
 805                                                 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
 806                                         }
 807                                         else {
 808                                                 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
 809                                         }
 810                                 }
 811                         }
 812                         else {
 813                                 if (k & 1) {
 814                                         if (j & 1) {
 815                                                 result = result + cln::factorial(n+k-1)
 816                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
 817                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
 818                                         }
 819                                         else {
 820                                                 result = result - cln::factorial(n+k-1)
 821                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
 822                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
 823                                         }
 824                                 }
 825                                 else {
 826                                         if (j & 1) {
 827                                                 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()
 828                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
 829                                         }
 830                                         else {
 831                                                 result = result + cln::factorial(n+k-1)
 832                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
 833                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
 834                                         }
 835                                 }
 836                         }
 837                 }
 838         }
 839         int np = n+p;
 840         if ((np-1) & 1) {
 841                 if (((np)/2+n) & 1) {
 842                         result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
 843                 }
 844                 else {
 845                         result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
 846                 }
 847         }
 848
 849         return result;
 850 }
 851
 852
 853 // helper function for S(n,p,x)
 854 // [Kol] remark to (9.1)
 855 cln::cl_N a_k(int k)
 856 {
 857         cln::cl_N result;
 858
 859         if (k == 0) {
 860                 return 1;
 861         }
 862
 863         result = result;
 864         for (int m=2; m<=k; m++) {
 865                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
 866         }
 867
 868         return -result / k;
 869 }
 870
 871
 872 // helper function for S(n,p,x)
 873 // [Kol] remark to (9.1)
 874 cln::cl_N b_k(int k)
 875 {
 876         cln::cl_N result;
 877
 878         if (k == 0) {
 879                 return 1;
 880         }
 881
 882         result = result;
 883         for (int m=2; m<=k; m++) {
 884                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
 885         }
 886
 887         return result / k;
 888 }
 889
 890
 891 // helper function for S(n,p,x)
 892 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
 893 {
 894         if (p==1) {
 895                 return Li_projection(n+1, x, prec);
 896         }
 897
 898         // check if precalculated values are sufficient
 899         if (p > ynsize+1) {
 900                 for (int i=ynsize; i<p-1; i++) {
 901                         fill_Yn(i, prec);
 902                 }
 903         }
 904
 905         // should be done otherwise
 906         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
 907         cln::cl_N xf = x * one;
 908         //cln::cl_N xf = x * cln::cl_float(1, prec);
 909
 910         cln::cl_N res;
 911         cln::cl_N resbuf;
 912         cln::cl_N factor = cln::expt(xf, p);
 913         int i = p;
 914         do {
 915                 resbuf = res;
 916                 if (i-p >= ynlength) {
 917                         // make Yn longer
 918                         make_Yn_longer(ynlength*2, prec);
 919                 }
 920                 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? ...
 921                 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
 922                 factor = factor * xf;
 923                 i++;
 924         } while (res != resbuf);
 925
 926         return res;
 927 }
 928
 929
 930 // helper function for S(n,p,x)
 931 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
 932 {
 933         // [Kol] (5.3)
 934         if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
 935
 936                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
 937                                    * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
 938
 939                 for (int s=0; s<n; s++) {
 940                         cln::cl_N res2;
 941                         for (int r=0; r<p; r++) {
 942                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
 943                                               * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
 944                         }
 945                         result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
 946                 }
 947
 948                 return result;
 949         }
 950
 951         return S_do_sum(n, p, x, prec);
 952 }
 953
 954
 955 // helper function for S(n,p,x)
 956 numeric S_num(int n, int p, const numeric& x)
 957 {
 958         if (x == 1) {
 959                 if (n == 1) {
 960                     // [Kol] (2.22) with (2.21)
 961                         return cln::zeta(p+1);
 962                 }
 963
 964                 if (p == 1) {
 965                     // [Kol] (2.22)
 966                         return cln::zeta(n+1);
 967                 }
 968
 969                 // [Kol] (9.1)
 970                 cln::cl_N result;
 971                 for (int nu=0; nu<n; nu++) {
 972                         for (int rho=0; rho<=p; rho++) {
 973                                 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
 974                                                   * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
 975                         }
 976                 }
 977                 result = result * cln::expt(cln::cl_I(-1),n+p-1);
 978
 979                 return result;
 980         }
 981         else if (x == -1) {
 982                 // [Kol] (2.22)
 983                 if (p == 1) {
 984                         return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
 985                 }
 986 //              throw std::runtime_error("don't know how to evaluate this function!");
 987         }
 988
 989         // what is the desired float format?
 990         // first guess: default format
 991         cln::float_format_t prec = cln::default_float_format;
 992         const cln::cl_N value = x.to_cl_N();
 993         // second guess: the argument's format
 994         if (!x.real().is_rational())
 995                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
 996         else if (!x.imag().is_rational())
 997                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
 998
 999         // [Kol] (5.3)
1000         if ((cln::realpart(value) < -0.5) || (n == 0)) {
1001
1002                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
1003                                    * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
1004
1005                 for (int s=0; s<n; s++) {
1006                         cln::cl_N res2;
1007                         for (int r=0; r<p; r++) {
1008                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
1009                                               * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
1010                         }
1011                         result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
1012                 }
1013
1014                 return result;
1015
1016         }
1017         // [Kol] (5.12)
1018         if (cln::abs(value) > 1) {
1019
1020                 cln::cl_N result;
1021
1022                 for (int s=0; s<p; s++) {
1023                         for (int r=0; r<=s; r++) {
1024                                 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
1025                                                   / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
1026                                                   * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
1027                         }
1028                 }
1029                 result = result * cln::expt(cln::cl_I(-1),n);
1030
1031                 cln::cl_N res2;
1032                 for (int r=0; r<n; r++) {
1033                         res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
1034                 }
1035                 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
1036
1037                 result = result + cln::expt(cln::cl_I(-1),p) * res2;
1038
1039                 return result;
1040         }
1041         else {
1042                 return S_projection(n, p, value, prec);
1043         }
1044 }
1045
1046
1047 } // end of anonymous namespace
1048
1049
1050 //////////////////////////////////////////////////////////////////////
1051 //
1052 // Nielsen's generalized polylogarithm  S(n,p,x)
1053 //
1054 // GiNaC function
1055 //
1056 //////////////////////////////////////////////////////////////////////
1057
1058
1059 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1060 {
1061         if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a<numeric>(x)) {
1062                 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1063         }
1064         return S(n, p, x).hold();
1065 }
1066
1067
1068 static ex S_eval(const ex& n, const ex& p, const ex& x)
1069 {
1070         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1071                 if (x == 0) {
1072                         return _ex0;
1073                 }
1074                 if (x == 1) {
1075                         lst m(n+1);
1076                         for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
1077                                 m.append(1);
1078                         }
1079                         return zeta(m);
1080                 }
1081                 if (p == 1) {
1082                         return Li(n+1, x);
1083                 }
1084                 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
1085                         return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1086                 }
1087         }
1088         if (n.is_zero()) {
1089                 // [Kol] (5.3)
1090                 return pow(-log(1-x), p) / factorial(p);
1091         }
1092         return S(n, p, x).hold();
1093 }
1094
1095
1096 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
1097 {
1098         epvector seq;
1099         seq.push_back(expair(S(n, p, x), 0));
1100         return pseries(rel, seq);
1101 }
1102
1103
1104 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
1105 {
1106         GINAC_ASSERT(deriv_param < 3);
1107         if (deriv_param < 2) {
1108                 return _ex0;
1109         }
1110         if (n > 0) {
1111                 return S(n-1, p, x) / x;
1112         } else {
1113                 return S(n, p-1, x) / (1-x);
1114         }
1115 }
1116
1117
1118 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
1119 {
1120         c.s << "\\mbox{S}_{";
1121         n.print(c);
1122         c.s << ",";
1123         p.print(c);
1124         c.s << "}(";
1125         x.print(c);
1126         c.s << ")";
1127 }
1128
1129
1130 REGISTER_FUNCTION(S,
1131                   evalf_func(S_evalf).
1132                   eval_func(S_eval).
1133                   series_func(S_series).
1134                   derivative_func(S_deriv).
1135                   print_func<print_latex>(S_print_latex).
1136                   do_not_evalf_params());
1137
1138
1139 //////////////////////////////////////////////////////////////////////
1140 //
1141 // Harmonic polylogarithm  H(m,x)
1142 //
1143 // helper functions
1144 //
1145 //////////////////////////////////////////////////////////////////////
1146
1147
1148 // anonymous namespace for helper functions
1149 namespace {
1150
1151
1152 // regulates the pole (used by 1/x-transformation)
1153 symbol H_polesign("IMSIGN");
1154
1155
1156 // convert parameters from H to Li representation
1157 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
1158 // returns true if some parameters are negative
1159 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
1160 {
1161         // expand parameter list
1162         lst mexp;
1163         for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
1164                 if (*it > 1) {
1165                         for (ex count=*it-1; count > 0; count--) {
1166                                 mexp.append(0);
1167                         }
1168                         mexp.append(1);
1169                 } else if (*it < -1) {
1170                         for (ex count=*it+1; count < 0; count++) {
1171                                 mexp.append(0);
1172                         }
1173                         mexp.append(-1);
1174                 } else {
1175                         mexp.append(*it);
1176                 }
1177         }
1178
1179         ex signum = 1;
1180         pf = 1;
1181         bool has_negative_parameters = false;
1182         ex acc = 1;
1183         for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
1184                 if (*it == 0) {
1185                         acc++;
1186                         continue;
1187                 }
1188                 if (*it > 0) {
1189                         m.append((*it+acc-1) * signum);
1190                 } else {
1191                         m.append((*it-acc+1) * signum);
1192                 }
1193                 acc = 1;
1194                 signum = *it;
1195                 pf *= *it;
1196                 if (pf < 0) {
1197                         has_negative_parameters = true;
1198                 }
1199         }
1200         if (has_negative_parameters) {
1201                 for (int i=0; i<m.nops(); i++) {
1202                         if (m.op(i) < 0) {
1203                                 m.let_op(i) = -m.op(i);
1204                                 s.append(-1);
1205                         } else {
1206                                 s.append(1);
1207                         }
1208                 }
1209         }
1210
1211         return has_negative_parameters;
1212 }
1213
1214
1215 // recursivly transforms H to corresponding multiple polylogarithms
1216 struct map_trafo_H_convert_to_Li : public map_function
1217 {
1218         ex operator()(const ex& e)
1219         {
1220                 if (is_a<add>(e) || is_a<mul>(e)) {
1221                         return e.map(*this);
1222                 }
1223                 if (is_a<function>(e)) {
1224                         std::string name = ex_to<function>(e).get_name();
1225                         if (name == "H") {
1226                                 lst parameter;
1227                                 if (is_a<lst>(e.op(0))) {
1228                                                 parameter = ex_to<lst>(e.op(0));
1229                                 } else {
1230                                         parameter = lst(e.op(0));
1231                                 }
1232                                 ex arg = e.op(1);
1233
1234                                 lst m;
1235                                 lst s;
1236                                 ex pf;
1237                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1238                                         s.let_op(0) = s.op(0) * arg;
1239                                         return pf * Li(m, s).hold();
1240                                 } else {
1241                                         for (int i=0; i<m.nops(); i++) {
1242                                                 s.append(1);
1243                                         }
1244                                         s.let_op(0) = s.op(0) * arg;
1245                                         return Li(m, s).hold();
1246                                 }
1247                         }
1248                 }
1249                 return e;
1250         }
1251 };
1252
1253
1254 // recursivly transforms H to corresponding zetas
1255 struct map_trafo_H_convert_to_zeta : public map_function
1256 {
1257         ex operator()(const ex& e)
1258         {
1259                 if (is_a<add>(e) || is_a<mul>(e)) {
1260                         return e.map(*this);
1261                 }
1262                 if (is_a<function>(e)) {
1263                         std::string name = ex_to<function>(e).get_name();
1264                         if (name == "H") {
1265                                 lst parameter;
1266                                 if (is_a<lst>(e.op(0))) {
1267                                                 parameter = ex_to<lst>(e.op(0));
1268                                 } else {
1269                                         parameter = lst(e.op(0));
1270                                 }
1271
1272                                 lst m;
1273                                 lst s;
1274                                 ex pf;
1275                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1276                                         return pf * zeta(m, s);
1277                                 } else {
1278                                         return zeta(m);
1279                                 }
1280                         }
1281                 }
1282                 return e;
1283         }
1284 };
1285
1286
1287 // remove trailing zeros from H-parameters
1288 struct map_trafo_H_reduce_trailing_zeros : public map_function
1289 {
1290         ex operator()(const ex& e)
1291         {
1292                 if (is_a<add>(e) || is_a<mul>(e)) {
1293                         return e.map(*this);
1294                 }
1295                 if (is_a<function>(e)) {
1296                         std::string name = ex_to<function>(e).get_name();
1297                         if (name == "H") {
1298                                 lst parameter;
1299                                 if (is_a<lst>(e.op(0))) {
1300                                         parameter = ex_to<lst>(e.op(0));
1301                                 } else {
1302                                         parameter = lst(e.op(0));
1303                                 }
1304                                 ex arg = e.op(1);
1305                                 if (parameter.op(parameter.nops()-1) == 0) {
1306
1307                                         //
1308                                         if (parameter.nops() == 1) {
1309                                                 return log(arg);
1310                                         }
1311
1312                                         //
1313                                         lst::const_iterator it = parameter.begin();
1314                                         while ((it != parameter.end()) && (*it == 0)) {
1315                                                 it++;
1316                                         }
1317                                         if (it == parameter.end()) {
1318                                                 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1319                                         }
1320
1321                                         //
1322                                         parameter.remove_last();
1323                                         int lastentry = parameter.nops();
1324                                         while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1325                                                 lastentry--;
1326                                         }
1327
1328                                         //
1329                                         ex result = log(arg) * H(parameter,arg).hold();
1330                                         ex acc = 0;
1331                                         for (ex i=0; i<lastentry; i++) {
1332                                                 if (parameter[i] > 0) {
1333                                                         parameter[i]++;
1334                                                         result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
1335                                                         parameter[i]--;
1336                                                         acc = 0;
1337                                                 } else if (parameter[i] < 0) {
1338                                                         parameter[i]--;
1339                                                         result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
1340                                                         parameter[i]++;
1341                                                         acc = 0;
1342                                                 } else {
1343                                                         acc++;
1344                                                 }
1345                                         }
1346
1347                                         if (lastentry < parameter.nops()) {
1348                                                 result = result / (parameter.nops()-lastentry+1);
1349                                                 return result.map(*this);
1350                                         } else {
1351                                                 return result;
1352                                         }
1353                                 }
1354                         }
1355                 }
1356                 return e;
1357         }
1358 };
1359
1360
1361 // returns an expression with zeta functions corresponding to the parameter list for H
1362 ex convert_H_to_zeta(const lst& m)
1363 {
1364         symbol xtemp("xtemp");
1365         map_trafo_H_reduce_trailing_zeros filter;
1366         map_trafo_H_convert_to_zeta filter2;
1367         return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
1368 }
1369
1370
1371 // convert signs form Li to H representation
1372 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
1373 {
1374         lst res;
1375         lst::const_iterator itm = m.begin();
1376         lst::const_iterator itx = ++x.begin();
1377         ex signum = _ex1;
1378         pf = _ex1;
1379         res.append(*itm);
1380         itm++;
1381         while (itx != x.end()) {
1382                 signum *= *itx;
1383                 pf *= signum;
1384                 res.append((*itm) * signum);
1385                 itm++;
1386                 itx++;
1387         }
1388         return res;
1389 }
1390
1391
1392 // multiplies an one-dimensional H with another H
1393 // [ReV] (18)
1394 ex trafo_H_mult(const ex& h1, const ex& h2)
1395 {
1396         ex res;
1397         ex hshort;
1398         lst hlong;
1399         ex h1nops = h1.op(0).nops();
1400         ex h2nops = h2.op(0).nops();
1401         if (h1nops > 1) {
1402                 hshort = h2.op(0).op(0);
1403                 hlong = ex_to<lst>(h1.op(0));
1404         } else {
1405                 hshort = h1.op(0).op(0);
1406                 if (h2nops > 1) {
1407                         hlong = ex_to<lst>(h2.op(0));
1408                 } else {
1409                         hlong = h2.op(0).op(0);
1410                 }
1411         }
1412         for (int i=0; i<=hlong.nops(); i++) {
1413                 lst newparameter;
1414                 int j=0;
1415                 for (; j<i; j++) {
1416                         newparameter.append(hlong[j]);
1417                 }
1418                 newparameter.append(hshort);
1419                 for (; j<hlong.nops(); j++) {
1420                         newparameter.append(hlong[j]);
1421                 }
1422                 res += H(newparameter, h1.op(1)).hold();
1423         }
1424         return res;
1425 }
1426
1427
1428 // applies trafo_H_mult recursively on expressions
1429 struct map_trafo_H_mult : public map_function
1430 {
1431         ex operator()(const ex& e)
1432         {
1433                 if (is_a<add>(e)) {
1434                         return e.map(*this);
1435                 }
1436
1437                 if (is_a<mul>(e)) {
1438
1439                         ex result = 1;
1440                         ex firstH;
1441                         lst Hlst;
1442                         for (int pos=0; pos<e.nops(); pos++) {
1443                                 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1444                                         std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1445                                         if (name == "H") {
1446                                                 for (ex i=0; i<e.op(pos).op(1); i++) {
1447                                                         Hlst.append(e.op(pos).op(0));
1448                                                 }
1449                                                 continue;
1450                                         }
1451                                 } else if (is_a<function>(e.op(pos))) {
1452                                         std::string name = ex_to<function>(e.op(pos)).get_name();
1453                                         if (name == "H") {
1454                                                 if (e.op(pos).op(0).nops() > 1) {
1455                                                         firstH = e.op(pos);
1456                                                 } else {
1457                                                         Hlst.append(e.op(pos));
1458                                                 }
1459                                                 continue;
1460                                         }
1461                                 }
1462                                 result *= e.op(pos);
1463                         }
1464                         if (firstH == 0) {
1465                                 if (Hlst.nops() > 0) {
1466                                         firstH = Hlst[Hlst.nops()-1];
1467                                         Hlst.remove_last();
1468                                 } else {
1469                                         return e;
1470                                 }
1471                         }
1472
1473                         if (Hlst.nops() > 0) {
1474                                 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1475                                 result *= buffer;
1476                                 for (int i=1; i<Hlst.nops(); i++) {
1477                                         result *= Hlst.op(i);
1478                                 }
1479                                 result = result.expand();
1480                                 map_trafo_H_mult recursion;
1481                                 return recursion(result);
1482                         } else {
1483                                 return e;
1484                         }
1485
1486                 }
1487                 return e;
1488         }
1489 };
1490
1491
1492 // do integration [ReV] (55)
1493 // put parameter 0 in front of existing parameters
1494 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
1495 {
1496         ex h;
1497         std::string name;
1498         if (is_a<function>(e)) {
1499                 name = ex_to<function>(e).get_name();
1500         }
1501         if (name == "H") {
1502                 h = e;
1503         } else {
1504                 for (int i=0; i<e.nops(); i++) {
1505                         if (is_a<function>(e.op(i))) {
1506                                 std::string name = ex_to<function>(e.op(i)).get_name();
1507                                 if (name == "H") {
1508                                         h = e.op(i);
1509                                 }
1510                         }
1511                 }
1512         }
1513         if (h != 0) {
1514                 lst newparameter = ex_to<lst>(h.op(0));
1515                 newparameter.prepend(0);
1516                 ex addzeta = convert_H_to_zeta(newparameter);
1517                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1518         } else {
1519                 return e * (-H(lst(0),1/arg).hold());
1520         }
1521 }
1522
1523
1524 // do integration [ReV] (55)
1525 // put parameter -1 in front of existing parameters
1526 ex trafo_H_1tx_prepend_minusone(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                 ex addzeta = convert_H_to_zeta(newparameter);
1549                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1550         } else {
1551                 ex addzeta = convert_H_to_zeta(lst(-1));
1552                 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
1553         }
1554 }
1555
1556
1557 // do integration [ReV] (55)
1558 // put parameter -1 in front of existing parameters
1559 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
1560 {
1561         ex h;
1562         std::string name;
1563         if (is_a<function>(e)) {
1564                 name = ex_to<function>(e).get_name();
1565         }
1566         if (name == "H") {
1567                 h = e;
1568         } else {
1569                 for (int i=0; i<e.nops(); i++) {
1570                         if (is_a<function>(e.op(i))) {
1571                                 std::string name = ex_to<function>(e.op(i)).get_name();
1572                                 if (name == "H") {
1573                                         h = e.op(i);
1574                                 }
1575                         }
1576                 }
1577         }
1578         if (h != 0) {
1579                 lst newparameter = ex_to<lst>(h.op(0));
1580                 newparameter.prepend(-1);
1581                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1582         } else {
1583                 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
1584         }
1585 }
1586
1587
1588 // do integration [ReV] (55)
1589 // put parameter 1 in front of existing parameters
1590 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
1591 {
1592         ex h;
1593         std::string name;
1594         if (is_a<function>(e)) {
1595                 name = ex_to<function>(e).get_name();
1596         }
1597         if (name == "H") {
1598                 h = e;
1599         } else {
1600                 for (int i=0; i<e.nops(); i++) {
1601                         if (is_a<function>(e.op(i))) {
1602                                 std::string name = ex_to<function>(e.op(i)).get_name();
1603                                 if (name == "H") {
1604                                         h = e.op(i);
1605                                 }
1606                         }
1607                 }
1608         }
1609         if (h != 0) {
1610                 lst newparameter = ex_to<lst>(h.op(0));
1611                 newparameter.prepend(1);
1612                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1613         } else {
1614                 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
1615         }
1616 }
1617
1618
1619 // do x -> 1/x transformation
1620 struct map_trafo_H_1overx : public map_function
1621 {
1622         ex operator()(const ex& e)
1623         {
1624                 if (is_a<add>(e) || is_a<mul>(e)) {
1625                         return e.map(*this);
1626                 }
1627
1628                 if (is_a<function>(e)) {
1629                         std::string name = ex_to<function>(e).get_name();
1630                         if (name == "H") {
1631
1632                                 lst parameter = ex_to<lst>(e.op(0));
1633                                 ex arg = e.op(1);
1634
1635                                 // special cases if all parameters are either 0, 1 or -1
1636                                 bool allthesame = true;
1637                                 if (parameter.op(0) == 0) {
1638                                         for (int i=1; i<parameter.nops(); i++) {
1639                                                 if (parameter.op(i) != 0) {
1640                                                         allthesame = false;
1641                                                         break;
1642                                                 }
1643                                         }
1644                                         if (allthesame) {
1645                                                 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1646                                         }
1647                                 } else if (parameter.op(0) == -1) {
1648                                         for (int i=1; i<parameter.nops(); i++) {
1649                                                 if (parameter.op(i) != -1) {
1650                                                         allthesame = false;
1651                                                         break;
1652                                                 }
1653                                         }
1654                                         if (allthesame) {
1655                                                 map_trafo_H_mult unify;
1656                                                 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
1657                                                        / factorial(parameter.nops())).expand());
1658                                         }
1659                                 } else {
1660                                         for (int i=1; i<parameter.nops(); i++) {
1661                                                 if (parameter.op(i) != 1) {
1662                                                         allthesame = false;
1663                                                         break;
1664                                                 }
1665                                         }
1666                                         if (allthesame) {
1667                                                 map_trafo_H_mult unify;
1668                                                 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
1669                                                        / factorial(parameter.nops())).expand());
1670                                         }
1671                                 }
1672
1673                                 lst newparameter = parameter;
1674                                 newparameter.remove_first();
1675
1676                                 if (parameter.op(0) == 0) {
1677
1678                                         // leading zero
1679                                         ex res = convert_H_to_zeta(parameter);
1680                                         map_trafo_H_1overx recursion;
1681                                         ex buffer = recursion(H(newparameter, arg).hold());
1682                                         if (is_a<add>(buffer)) {
1683                                                 for (int i=0; i<buffer.nops(); i++) {
1684                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
1685                                                 }
1686                                         } else {
1687                                                 res += trafo_H_1tx_prepend_zero(buffer, arg);
1688                                         }
1689                                         return res;
1690
1691                                 } else if (parameter.op(0) == -1) {
1692
1693                                         // leading negative one
1694                                         ex res = convert_H_to_zeta(parameter);
1695                                         map_trafo_H_1overx recursion;
1696                                         ex buffer = recursion(H(newparameter, arg).hold());
1697                                         if (is_a<add>(buffer)) {
1698                                                 for (int i=0; i<buffer.nops(); i++) {
1699                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
1700                                                 }
1701                                         } else {
1702                                                 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
1703                                         }
1704                                         return res;
1705
1706                                 } else {
1707
1708                                         // leading one
1709                                         map_trafo_H_1overx recursion;
1710                                         map_trafo_H_mult unify;
1711                                         ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1712                                         int firstzero = 0;
1713                                         while (parameter.op(firstzero) == 1) {
1714                                                 firstzero++;
1715                                         }
1716                                         for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1717                                                 lst newparameter;
1718                                                 int j=0;
1719                                                 for (; j<=i; j++) {
1720                                                         newparameter.append(parameter[j+1]);
1721                                                 }
1722                                                 newparameter.append(1);
1723                                                 for (; j<parameter.nops()-1; j++) {
1724                                                         newparameter.append(parameter[j+1]);
1725                                                 }
1726                                                 res -= H(newparameter, arg).hold();
1727                                         }
1728                                         res = recursion(res).expand() / firstzero;
1729                                         return unify(res);
1730
1731                                 }
1732
1733                         }
1734                 }
1735                 return e;
1736         }
1737 };
1738
1739
1740 // do x -> (1-x)/(1+x) transformation
1741 struct map_trafo_H_1mxt1px : public map_function
1742 {
1743         ex operator()(const ex& e)
1744         {
1745                 if (is_a<add>(e) || is_a<mul>(e)) {
1746                         return e.map(*this);
1747                 }
1748
1749                 if (is_a<function>(e)) {
1750                         std::string name = ex_to<function>(e).get_name();
1751                         if (name == "H") {
1752
1753                                 lst parameter = ex_to<lst>(e.op(0));
1754                                 ex arg = e.op(1);
1755
1756                                 // special cases if all parameters are either 0, 1 or -1
1757                                 bool allthesame = true;
1758                                 if (parameter.op(0) == 0) {
1759                                         for (int i=1; i<parameter.nops(); i++) {
1760                                                 if (parameter.op(i) != 0) {
1761                                                         allthesame = false;
1762                                                         break;
1763                                                 }
1764                                         }
1765                                         if (allthesame) {
1766                                                 map_trafo_H_mult unify;
1767                                                 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1768                                                        / factorial(parameter.nops())).expand());
1769                                         }
1770                                 } else if (parameter.op(0) == -1) {
1771                                         for (int i=1; i<parameter.nops(); i++) {
1772                                                 if (parameter.op(i) != -1) {
1773                                                         allthesame = false;
1774                                                         break;
1775                                                 }
1776                                         }
1777                                         if (allthesame) {
1778                                                 map_trafo_H_mult unify;
1779                                                 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1780                                                        / factorial(parameter.nops())).expand());
1781                                         }
1782                                 } else {
1783                                         for (int i=1; i<parameter.nops(); i++) {
1784                                                 if (parameter.op(i) != 1) {
1785                                                         allthesame = false;
1786                                                         break;
1787                                                 }
1788                                         }
1789                                         if (allthesame) {
1790                                                 map_trafo_H_mult unify;
1791                                                 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1792                                                        / factorial(parameter.nops())).expand());
1793                                         }
1794                                 }
1795
1796                                 lst newparameter = parameter;
1797                                 newparameter.remove_first();
1798
1799                                 if (parameter.op(0) == 0) {
1800
1801                                         // leading zero
1802                                         ex res = convert_H_to_zeta(parameter);
1803                                         map_trafo_H_1mxt1px recursion;
1804                                         ex buffer = recursion(H(newparameter, arg).hold());
1805                                         if (is_a<add>(buffer)) {
1806                                                 for (int i=0; i<buffer.nops(); i++) {
1807                                                         res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1808                                                 }
1809                                         } else {
1810                                                 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1811                                         }
1812                                         return res;
1813
1814                                 } else if (parameter.op(0) == -1) {
1815
1816                                         // leading negative one
1817                                         ex res = convert_H_to_zeta(parameter);
1818                                         map_trafo_H_1mxt1px recursion;
1819                                         ex buffer = recursion(H(newparameter, arg).hold());
1820                                         if (is_a<add>(buffer)) {
1821                                                 for (int i=0; i<buffer.nops(); i++) {
1822                                                         res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1823                                                 }
1824                                         } else {
1825                                                 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1826                                         }
1827                                         return res;
1828
1829                                 } else {
1830
1831                                         // leading one
1832                                         map_trafo_H_1mxt1px recursion;
1833                                         map_trafo_H_mult unify;
1834                                         ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1835                                         int firstzero = 0;
1836                                         while (parameter.op(firstzero) == 1) {
1837                                                 firstzero++;
1838                                         }
1839                                         for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1840                                                 lst newparameter;
1841                                                 int j=0;
1842                                                 for (; j<=i; j++) {
1843                                                         newparameter.append(parameter[j+1]);
1844                                                 }
1845                                                 newparameter.append(1);
1846                                                 for (; j<parameter.nops()-1; j++) {
1847                                                         newparameter.append(parameter[j+1]);
1848                                                 }
1849                                                 res -= H(newparameter, arg).hold();
1850                                         }
1851                                         res = recursion(res).expand() / firstzero;
1852                                         return unify(res);
1853
1854                                 }
1855
1856                         }
1857                 }
1858                 return e;
1859         }
1860 };
1861
1862
1863 // do the actual summation.
1864 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1865 {
1866         const int j = m.size();
1867
1868         std::vector<cln::cl_N> t(j);
1869
1870         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1871         cln::cl_N factor = cln::expt(x, j) * one;
1872         cln::cl_N t0buf;
1873         int q = 0;
1874         do {
1875                 t0buf = t[0];
1876                 q++;
1877                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1878                 for (int k=j-2; k>=1; k--) {
1879                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1880                 }
1881                 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1882                 factor = factor * x;
1883         } while (t[0] != t0buf);
1884
1885         return t[0];
1886 }
1887
1888
1889 } // end of anonymous namespace
1890
1891
1892 //////////////////////////////////////////////////////////////////////
1893 //
1894 // Harmonic polylogarithm  H(m,x)
1895 //
1896 // GiNaC function
1897 //
1898 //////////////////////////////////////////////////////////////////////
1899
1900
1901 static ex H_evalf(const ex& x1, const ex& x2)
1902 {
1903         if (is_a<lst>(x1) && is_a<numeric>(x2)) {
1904                 for (int i=0; i<x1.nops(); i++) {
1905                         if (!x1.op(i).info(info_flags::integer)) {
1906                                 return H(x1,x2).hold();
1907                         }
1908                 }
1909                 if (x1.nops() < 1) {
1910                         return H(x1,x2).hold();
1911                 }
1912
1913                 cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
1914
1915                 const lst& morg = ex_to<lst>(x1);
1916                 // remove trailing zeros ...
1917                 if (*(--morg.end()) == 0) {
1918                         symbol xtemp("xtemp");
1919                         map_trafo_H_reduce_trailing_zeros filter;
1920                         return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
1921                 }
1922                 // ... and expand parameter notation
1923                 lst m;
1924                 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
1925                         if (*it > 1) {
1926                                 for (ex count=*it-1; count > 0; count--) {
1927                                         m.append(0);
1928                                 }
1929                                 m.append(1);
1930                         } else if (*it < -1) {
1931                                 for (ex count=*it+1; count < 0; count++) {
1932                                         m.append(0);
1933                                 }
1934                                 m.append(-1);
1935                         } else {
1936                                 m.append(*it);
1937                         }
1938                 }
1939
1940                 // since the transformations produce a lot of terms, they are only efficient for
1941                 // argument near one.
1942                 // no transformation needed -> do summation
1943                 if (cln::abs(x) < 0.95) {
1944                         lst m_lst;
1945                         lst s_lst;
1946                         ex pf;
1947                         if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
1948                                 // negative parameters -> s_lst is filled
1949                                 std::vector<int> m_int;
1950                                 std::vector<cln::cl_N> x_cln;
1951                                 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
1952                                      it_int != m_lst.end(); it_int++, it_cln++) {
1953                                         m_int.push_back(ex_to<numeric>(*it_int).to_int());
1954                                         x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
1955                                 }
1956                                 x_cln.front() = x_cln.front() * x;
1957                                 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
1958                         } else {
1959                                 // only positive parameters
1960                                 //TODO
1961                                 if (m_lst.nops() == 1) {
1962                                         return Li(m_lst.op(0), x2).evalf();
1963                                 }
1964                                 std::vector<int> m_int;
1965                                 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
1966                                         m_int.push_back(ex_to<numeric>(*it).to_int());
1967                                 }
1968                                 return numeric(H_do_sum(m_int, x));
1969                         }
1970                 }
1971
1972                 ex res = 1;
1973
1974                 // ensure that the realpart of the argument is positive
1975                 if (cln::realpart(x) < 0) {
1976                         x = -x;
1977                         for (int i=0; i<m.nops(); i++) {
1978                                 if (m.op(i) != 0) {
1979                                         m.let_op(i) = -m.op(i);
1980                                         res *= -1;
1981                                 }
1982                         }
1983                 }
1984
1985                 // choose transformations
1986                 symbol xtemp("xtemp");
1987                 if (cln::abs(x-1) < 1.4142) {
1988                         // x -> (1-x)/(1+x)
1989                         map_trafo_H_1mxt1px trafo;
1990                         res *= trafo(H(m, xtemp));
1991                 } else {
1992                         // x -> 1/x
1993                         map_trafo_H_1overx trafo;
1994                         res *= trafo(H(m, xtemp));
1995                         if (cln::imagpart(x) <= 0) {
1996                                 res = res.subs(H_polesign == -I*Pi);
1997                         } else {
1998                                 res = res.subs(H_polesign == I*Pi);
1999                         }
2000                 }
2001
2002                 // simplify result
2003 // TODO
2004 //              map_trafo_H_convert converter;
2005 //              res = converter(res).expand();
2006 //              lst ll;
2007 //              res.find(H(wild(1),wild(2)), ll);
2008 //              res.find(zeta(wild(1)), ll);
2009 //              res.find(zeta(wild(1),wild(2)), ll);
2010 //              res = res.collect(ll);
2011
2012                 return res.subs(xtemp == numeric(x)).evalf();
2013         }
2014
2015         return H(x1,x2).hold();
2016 }
2017
2018
2019 static ex H_eval(const ex& m_, const ex& x)
2020 {
2021         lst m;
2022         if (is_a<lst>(m_)) {
2023                 m = ex_to<lst>(m_);
2024         } else {
2025                 m = lst(m_);
2026         }
2027         if (m.nops() == 0) {
2028                 return _ex1;
2029         }
2030         ex pos1;
2031         ex pos2;
2032         ex n;
2033         ex p;
2034         int step = 0;
2035         if (*m.begin() > _ex1) {
2036                 step++;
2037                 pos1 = _ex0;
2038                 pos2 = _ex1;
2039                 n = *m.begin()-1;
2040                 p = _ex1;
2041         } else if (*m.begin() < _ex_1) {
2042                 step++;
2043                 pos1 = _ex0;
2044                 pos2 = _ex_1;
2045                 n = -*m.begin()-1;
2046                 p = _ex1;
2047         } else if (*m.begin() == _ex0) {
2048                 pos1 = _ex0;
2049                 n = _ex1;
2050         } else {
2051                 pos1 = *m.begin();
2052                 p = _ex1;
2053         }
2054         for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
2055                 if ((*it).info(info_flags::integer)) {
2056                         if (step == 0) {
2057                                 if (*it > _ex1) {
2058                                         if (pos1 == _ex0) {
2059                                                 step = 1;
2060                                                 pos2 = _ex1;
2061                                                 n += *it-1;
2062                                                 p = _ex1;
2063                                         } else {
2064                                                 step = 2;
2065                                         }
2066                                 } else if (*it < _ex_1) {
2067                                         if (pos1 == _ex0) {
2068                                                 step = 1;
2069                                                 pos2 = _ex_1;
2070                                                 n += -*it-1;
2071                                                 p = _ex1;
2072                                         } else {
2073                                                 step = 2;
2074                                         }
2075                                 } else {
2076                                         if (*it != pos1) {
2077                                                 step = 1;
2078                                                 pos2 = *it;
2079                                         }
2080                                         if (*it == _ex0) {
2081                                                 n++;
2082                                         } else {
2083                                                 p++;
2084                                         }
2085                                 }
2086                         } else if (step == 1) {
2087                                 if (*it != pos2) {
2088                                         step = 2;
2089                                 } else {
2090                                         if (*it == _ex0) {
2091                                                 n++;
2092                                         } else {
2093                                                 p++;
2094                                         }
2095                                 }
2096                         }
2097                 } else {
2098                         // if some m_i is not an integer
2099                         return H(m_, x).hold();
2100                 }
2101         }
2102         if ((x == _ex1) && (*(--m.end()) != _ex0)) {
2103                 return convert_H_to_zeta(m);
2104         }
2105         if (step == 0) {
2106                 if (pos1 == _ex0) {
2107                         // all zero
2108                         if (x == _ex0) {
2109                                 return H(m_, x).hold();
2110                         }
2111                         return pow(log(x), m.nops()) / factorial(m.nops());
2112                 } else {
2113                         // all (minus) one
2114                         return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
2115                 }
2116         } else if ((step == 1) && (pos1 == _ex0)){
2117                 // convertible to S
2118                 if (pos2 == _ex1) {
2119                         return S(n, p, x);
2120                 } else {
2121                         return pow(-1, p) * S(n, p, -x);
2122                 }
2123         }
2124         if (x == _ex0) {
2125                 return _ex0;
2126         }
2127         if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2128                 return H(m_, x).evalf();
2129         }
2130         return H(m_, x).hold();
2131 }
2132
2133
2134 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
2135 {
2136         epvector seq;
2137         seq.push_back(expair(H(m, x), 0));
2138         return pseries(rel, seq);
2139 }
2140
2141
2142 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
2143 {
2144         GINAC_ASSERT(deriv_param < 2);
2145         if (deriv_param == 0) {
2146                 return _ex0;
2147         }
2148         lst m;
2149         if (is_a<lst>(m_)) {
2150                 m = ex_to<lst>(m_);
2151         } else {
2152                 m = lst(m_);
2153         }
2154         ex mb = *m.begin();
2155         if (mb > _ex1) {
2156                 m[0]--;
2157                 return H(m, x) / x;
2158         }
2159         if (mb < _ex_1) {
2160                 m[0]++;
2161                 return H(m, x) / x;
2162         }
2163         m.remove_first();
2164         if (mb == _ex1) {
2165                 return 1/(1-x) * H(m, x);
2166         } else if (mb == _ex_1) {
2167                 return 1/(1+x) * H(m, x);
2168         } else {
2169                 return H(m, x) / x;
2170         }
2171 }
2172
2173
2174 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
2175 {
2176         lst m;
2177         if (is_a<lst>(m_)) {
2178                 m = ex_to<lst>(m_);
2179         } else {
2180                 m = lst(m_);
2181         }
2182         c.s << "\\mbox{H}_{";
2183         lst::const_iterator itm = m.begin();
2184         (*itm).print(c);
2185         itm++;
2186         for (; itm != m.end(); itm++) {
2187                 c.s << ",";
2188                 (*itm).print(c);
2189         }
2190         c.s << "}(";
2191         x.print(c);
2192         c.s << ")";
2193 }
2194
2195
2196 REGISTER_FUNCTION(H,
2197                   evalf_func(H_evalf).
2198                   eval_func(H_eval).
2199                   series_func(H_series).
2200                   derivative_func(H_deriv).
2201                   print_func<print_latex>(H_print_latex).
2202                   do_not_evalf_params());
2203
2204
2205 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
2206 ex convert_H_to_Li(const ex& m, const ex& x)
2207 {
2208         map_trafo_H_reduce_trailing_zeros filter;
2209         map_trafo_H_convert_to_Li filter2;
2210         if (is_a<lst>(m)) {
2211                 return filter2(filter(H(m, x).hold()));
2212         } else {
2213                 return filter2(filter(H(lst(m), x).hold()));
2214         }
2215 }
2216
2217
2218 //////////////////////////////////////////////////////////////////////
2219 //
2220 // Multiple zeta values  zeta(x) and zeta(x,s)
2221 //
2222 // helper functions
2223 //
2224 //////////////////////////////////////////////////////////////////////
2225
2226
2227 // anonymous namespace for helper functions
2228 namespace {
2229
2230
2231 // parameters and data for [Cra] algorithm
2232 const cln::cl_N lambda = cln::cl_N("319/320");
2233 int L1;
2234 int L2;
2235 std::vector<std::vector<cln::cl_N> > f_kj;
2236 std::vector<cln::cl_N> crB;
2237 std::vector<std::vector<cln::cl_N> > crG;
2238 std::vector<cln::cl_N> crX;
2239
2240
2241 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
2242 {
2243         const int size = a.size();
2244         for (int n=0; n<size; n++) {
2245                 c[n] = 0;
2246                 for (int m=0; m<=n; m++) {
2247                         c[n] = c[n] + a[m]*b[n-m];
2248                 }
2249         }
2250 }
2251
2252
2253 // [Cra] section 4
2254 void initcX(const std::vector<int>& s)
2255 {
2256         const int k = s.size();
2257
2258         crX.clear();
2259         crG.clear();
2260         crB.clear();
2261
2262         for (int i=0; i<=L2; i++) {
2263                 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
2264         }
2265
2266         int Sm = 0;
2267         int Smp1 = 0;
2268         for (int m=0; m<k-1; m++) {
2269                 std::vector<cln::cl_N> crGbuf;
2270                 Sm = Sm + s[m];
2271                 Smp1 = Sm + s[m+1];
2272                 for (int i=0; i<=L2; i++) {
2273                         crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
2274                 }
2275                 crG.push_back(crGbuf);
2276         }
2277
2278         crX = crB;
2279
2280         for (int m=0; m<k-1; m++) {
2281                 std::vector<cln::cl_N> Xbuf;
2282                 for (int i=0; i<=L2; i++) {
2283                         Xbuf.push_back(crX[i] * crG[m][i]);
2284                 }
2285                 halfcyclic_convolute(Xbuf, crB, crX);
2286         }
2287 }
2288
2289
2290 // [Cra] section 4
2291 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
2292 {
2293         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2294         cln::cl_N factor = cln::expt(lambda, Sqk);
2295         cln::cl_N res = factor / Sqk * crX[0] * one;
2296         cln::cl_N resbuf;
2297         int N = 0;
2298         do {
2299                 resbuf = res;
2300                 factor = factor * lambda;
2301                 N++;
2302                 res = res + crX[N] * factor / (N+Sqk);
2303         } while ((res != resbuf) || cln::zerop(crX[N]));
2304         return res;
2305 }
2306
2307
2308 // [Cra] section 4
2309 void calc_f(int maxr)
2310 {
2311         f_kj.clear();
2312         f_kj.resize(L1);
2313
2314         cln::cl_N t0, t1, t2, t3, t4;
2315         int i, j, k;
2316         std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
2317         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2318
2319         t0 = cln::exp(-lambda);
2320         t2 = 1;
2321         for (k=1; k<=L1; k++) {
2322                 t1 = k * lambda;
2323                 t2 = t0 * t2;
2324                 for (j=1; j<=maxr; j++) {
2325                         t3 = 1;
2326                         t4 = 1;
2327                         for (i=2; i<=j; i++) {
2328                                 t4 = t4 * (j-i+1);
2329                                 t3 = t1 * t3 + t4;
2330                         }
2331                         (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
2332                 }
2333                 it++;
2334         }
2335 }
2336
2337
2338 // [Cra] (3.1)
2339 cln::cl_N crandall_Z(const std::vector<int>& s)
2340 {
2341         const int j = s.size();
2342
2343         if (j == 1) {
2344                 cln::cl_N t0;
2345                 cln::cl_N t0buf;
2346                 int q = 0;
2347                 do {
2348                         t0buf = t0;
2349                         q++;
2350                         t0 = t0 + f_kj[q+j-2][s[0]-1];
2351                 } while (t0 != t0buf);
2352
2353                 return t0 / cln::factorial(s[0]-1);
2354         }
2355
2356         std::vector<cln::cl_N> t(j);
2357
2358         cln::cl_N t0buf;
2359         int q = 0;
2360         do {
2361                 t0buf = t[0];
2362                 q++;
2363                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
2364                 for (int k=j-2; k>=1; k--) {
2365                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
2366                 }
2367                 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
2368         } while (t[0] != t0buf);
2369
2370         return t[0] / cln::factorial(s[0]-1);
2371 }
2372
2373
2374 // [Cra] (2.4)
2375 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
2376 {
2377         std::vector<int> r = s;
2378         const int j = r.size();
2379
2380         // decide on maximal size of f_kj for crandall_Z
2381         if (Digits < 50) {
2382                 L1 = 150;
2383         } else {
2384                 L1 = Digits * 3 + j*2;
2385         }
2386
2387         // decide on maximal size of crX for crandall_Y
2388         if (Digits < 38) {
2389                 L2 = 63;
2390         } else if (Digits < 86) {
2391                 L2 = 127;
2392         } else if (Digits < 192) {
2393                 L2 = 255;
2394         } else if (Digits < 394) {
2395                 L2 = 511;
2396         } else if (Digits < 808) {
2397                 L2 = 1023;
2398         } else {
2399                 L2 = 2047;
2400         }
2401
2402         cln::cl_N res;
2403
2404         int maxr = 0;
2405         int S = 0;
2406         for (int i=0; i<j; i++) {
2407                 S += r[i];
2408                 if (r[i] > maxr) {
2409                         maxr = r[i];
2410                 }
2411         }
2412
2413         calc_f(maxr);
2414
2415         const cln::cl_N r0factorial = cln::factorial(r[0]-1);
2416
2417         std::vector<int> rz;
2418         int skp1buf;
2419         int Srun = S;
2420         for (int k=r.size()-1; k>0; k--) {
2421
2422                 rz.insert(rz.begin(), r.back());
2423                 skp1buf = rz.front();
2424                 Srun -= skp1buf;
2425                 r.pop_back();
2426
2427                 initcX(r);
2428
2429                 for (int q=0; q<skp1buf; q++) {
2430
2431                         cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2432                         cln::cl_N pp2 = crandall_Z(rz);
2433
2434                         rz.front()--;
2435
2436                         if (q & 1) {
2437                                 res = res - pp1 * pp2 / cln::factorial(q);
2438                         } else {
2439                                 res = res + pp1 * pp2 / cln::factorial(q);
2440                         }
2441                 }
2442                 rz.front() = skp1buf;
2443         }
2444         rz.insert(rz.begin(), r.back());
2445
2446         initcX(rz);
2447
2448         res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2449
2450         return res;
2451 }
2452
2453
2454 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2455 {
2456         const int j = r.size();
2457
2458         // buffer for subsums
2459         std::vector<cln::cl_N> t(j);
2460         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2461
2462         cln::cl_N t0buf;
2463         int q = 0;
2464         do {
2465                 t0buf = t[0];
2466                 q++;
2467                 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2468                 for (int k=j-2; k>=0; k--) {
2469                         t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2470                 }
2471         } while (t[0] != t0buf);
2472
2473         return t[0];
2474 }
2475
2476
2477 // does Hoelder convolution. see [BBB] (7.0)
2478 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2479 {
2480         // prepare parameters
2481         // holds Li arguments in [BBB] notation
2482         std::vector<int> s = s_;
2483         std::vector<int> m_p = m_;
2484         std::vector<int> m_q;
2485         // holds Li arguments in nested sums notation
2486         std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2487         s_p[0] = s_p[0] * cln::cl_N("1/2");
2488         // convert notations
2489         int sig = 1;
2490         for (int i=0; i<s_.size(); i++) {
2491                 if (s_[i] < 0) {
2492                         sig = -sig;
2493                         s_p[i] = -s_p[i];
2494                 }
2495                 s[i] = sig * std::abs(s[i]);
2496         }
2497         std::vector<cln::cl_N> s_q;
2498         cln::cl_N signum = 1;
2499
2500         // first term
2501         cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2502
2503         // middle terms
2504         do {
2505
2506                 // change parameters
2507                 if (s.front() > 0) {
2508                         if (m_p.front() == 1) {
2509                                 m_p.erase(m_p.begin());
2510                                 s_p.erase(s_p.begin());
2511                                 if (s_p.size() > 0) {
2512                                         s_p.front() = s_p.front() * cln::cl_N("1/2");
2513                                 }
2514                                 s.erase(s.begin());
2515                                 m_q.front()++;
2516                         } else {
2517                                 m_p.front()--;
2518                                 m_q.insert(m_q.begin(), 1);
2519                                 if (s_q.size() > 0) {
2520                                         s_q.front() = s_q.front() * 2;
2521                                 }
2522                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2523                         }
2524                 } else {
2525                         if (m_p.front() == 1) {
2526                                 m_p.erase(m_p.begin());
2527                                 cln::cl_N spbuf = s_p.front();
2528                                 s_p.erase(s_p.begin());
2529                                 if (s_p.size() > 0) {
2530                                         s_p.front() = s_p.front() * spbuf;
2531                                 }
2532                                 s.erase(s.begin());
2533                                 m_q.insert(m_q.begin(), 1);
2534                                 if (s_q.size() > 0) {
2535                                         s_q.front() = s_q.front() * 4;
2536                                 }
2537                                 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2538                                 signum = -signum;
2539                         } else {
2540                                 m_p.front()--;
2541                                 m_q.insert(m_q.begin(), 1);
2542                                 if (s_q.size() > 0) {
2543                                         s_q.front() = s_q.front() * 2;
2544                                 }
2545                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2546                         }
2547                 }
2548
2549                 // exiting the loop
2550                 if (m_p.size() == 0) break;
2551
2552                 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2553
2554         } while (true);
2555
2556         // last term
2557         res = res + signum * multipleLi_do_sum(m_q, s_q);
2558
2559         return res;
2560 }
2561
2562
2563 } // end of anonymous namespace
2564
2565
2566 //////////////////////////////////////////////////////////////////////
2567 //
2568 // Multiple zeta values  zeta(x)
2569 //
2570 // GiNaC function
2571 //
2572 //////////////////////////////////////////////////////////////////////
2573
2574
2575 static ex zeta1_evalf(const ex& x)
2576 {
2577         if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2578
2579                 // multiple zeta value
2580                 const int count = x.nops();
2581                 const lst& xlst = ex_to<lst>(x);
2582                 std::vector<int> r(count);
2583
2584                 // check parameters and convert them
2585                 lst::const_iterator it1 = xlst.begin();
2586                 std::vector<int>::iterator it2 = r.begin();
2587                 do {
2588                         if (!(*it1).info(info_flags::posint)) {
2589                                 return zeta(x).hold();
2590                         }
2591                         *it2 = ex_to<numeric>(*it1).to_int();
2592                         it1++;
2593                         it2++;
2594                 } while (it2 != r.end());
2595
2596                 // check for divergence
2597                 if (r[0] == 1) {
2598                         return zeta(x).hold();
2599                 }
2600
2601                 // decide on summation algorithm
2602                 // this is still a bit clumsy
2603                 int limit = (Digits>17) ? 10 : 6;
2604                 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2605                         return numeric(zeta_do_sum_Crandall(r));
2606                 } else {
2607                         return numeric(zeta_do_sum_simple(r));
2608                 }
2609         }
2610
2611         // single zeta value
2612         if (is_exactly_a<numeric>(x) && (x != 1)) {
2613                 try {
2614                         return zeta(ex_to<numeric>(x));
2615                 } catch (const dunno &e) { }
2616         }
2617
2618         return zeta(x).hold();
2619 }
2620
2621
2622 static ex zeta1_eval(const ex& m)
2623 {
2624         if (is_exactly_a<lst>(m)) {
2625                 if (m.nops() == 1) {
2626                         return zeta(m.op(0));
2627                 }
2628                 return zeta(m).hold();
2629         }
2630
2631         if (m.info(info_flags::numeric)) {
2632                 const numeric& y = ex_to<numeric>(m);
2633                 // trap integer arguments:
2634                 if (y.is_integer()) {
2635                         if (y.is_zero()) {
2636                                 return _ex_1_2;
2637                         }
2638                         if (y.is_equal(_num1)) {
2639                                 return zeta(m).hold();
2640                         }
2641                         if (y.info(info_flags::posint)) {
2642                                 if (y.info(info_flags::odd)) {
2643                                         return zeta(m).hold();
2644                                 } else {
2645                                         return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2646                                 }
2647                         } else {
2648                                 if (y.info(info_flags::odd)) {
2649                                         return -bernoulli(_num1-y) / (_num1-y);
2650                                 } else {
2651                                         return _ex0;
2652                                 }
2653                         }
2654                 }
2655                 // zeta(float)
2656                 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
2657                         return zeta1_evalf(m);
2658                 }
2659         }
2660         return zeta(m).hold();
2661 }
2662
2663
2664 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
2665 {
2666         GINAC_ASSERT(deriv_param==0);
2667
2668         if (is_exactly_a<lst>(m)) {
2669                 return _ex0;
2670         } else {
2671                 return zetaderiv(_ex1, m);
2672         }
2673 }
2674
2675
2676 static void zeta1_print_latex(const ex& m_, const print_context& c)
2677 {
2678         c.s << "\\zeta(";
2679         if (is_a<lst>(m_)) {
2680                 const lst& m = ex_to<lst>(m_);
2681                 lst::const_iterator it = m.begin();
2682                 (*it).print(c);
2683                 it++;
2684                 for (; it != m.end(); it++) {
2685                         c.s << ",";
2686                         (*it).print(c);
2687                 }
2688         } else {
2689                 m_.print(c);
2690         }
2691         c.s << ")";
2692 }
2693
2694
2695 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
2696                                 evalf_func(zeta1_evalf).
2697                                 eval_func(zeta1_eval).
2698                                 derivative_func(zeta1_deriv).
2699                                 print_func<print_latex>(zeta1_print_latex).
2700                                 do_not_evalf_params().
2701                                 overloaded(2));
2702
2703
2704 //////////////////////////////////////////////////////////////////////
2705 //
2706 // Alternating Euler sum  zeta(x,s)
2707 //
2708 // GiNaC function
2709 //
2710 //////////////////////////////////////////////////////////////////////
2711
2712
2713 static ex zeta2_evalf(const ex& x, const ex& s)
2714 {
2715         if (is_exactly_a<lst>(x)) {
2716
2717                 // alternating Euler sum
2718                 const int count = x.nops();
2719                 const lst& xlst = ex_to<lst>(x);
2720                 const lst& slst = ex_to<lst>(s);
2721                 std::vector<int> xi(count);
2722                 std::vector<int> si(count);
2723
2724                 // check parameters and convert them
2725                 lst::const_iterator it_xread = xlst.begin();
2726                 lst::const_iterator it_sread = slst.begin();
2727                 std::vector<int>::iterator it_xwrite = xi.begin();
2728                 std::vector<int>::iterator it_swrite = si.begin();
2729                 do {
2730                         if (!(*it_xread).info(info_flags::posint)) {
2731                                 return zeta(x, s).hold();
2732                         }
2733                         *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2734                         if (*it_sread > 0) {
2735                                 *it_swrite = 1;
2736                         } else {
2737                                 *it_swrite = -1;
2738                         }
2739                         it_xread++;
2740                         it_sread++;
2741                         it_xwrite++;
2742                         it_swrite++;
2743                 } while (it_xwrite != xi.end());
2744
2745                 // check for divergence
2746                 if ((xi[0] == 1) && (si[0] == 1)) {
2747                         return zeta(x, s).hold();
2748                 }
2749
2750                 // use Hoelder convolution
2751                 return numeric(zeta_do_Hoelder_convolution(xi, si));
2752         }
2753
2754         return zeta(x, s).hold();
2755 }
2756
2757
2758 static ex zeta2_eval(const ex& m, const ex& s_)
2759 {
2760         if (is_exactly_a<lst>(s_)) {
2761                 const lst& s = ex_to<lst>(s_);
2762                 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
2763                         if ((*it).info(info_flags::positive)) {
2764                                 continue;
2765                         }
2766                         return zeta(m, s_).hold();
2767                 }
2768                 return zeta(m);
2769         } else if (s_.info(info_flags::positive)) {
2770                 return zeta(m);
2771         }
2772
2773         return zeta(m, s_).hold();
2774 }
2775
2776
2777 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
2778 {
2779         GINAC_ASSERT(deriv_param==0);
2780
2781         if (is_exactly_a<lst>(m)) {
2782                 return _ex0;
2783         } else {
2784                 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
2785                         return zetaderiv(_ex1, m);
2786                 }
2787                 return _ex0;
2788         }
2789 }
2790
2791
2792 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
2793 {
2794         lst m;
2795         if (is_a<lst>(m_)) {
2796                 m = ex_to<lst>(m_);
2797         } else {
2798                 m = lst(m_);
2799         }
2800         lst s;
2801         if (is_a<lst>(s_)) {
2802                 s = ex_to<lst>(s_);
2803         } else {
2804                 s = lst(s_);
2805         }
2806         c.s << "\\zeta(";
2807         lst::const_iterator itm = m.begin();
2808         lst::const_iterator its = s.begin();
2809         if (*its < 0) {
2810                 c.s << "\\overline{";
2811                 (*itm).print(c);
2812                 c.s << "}";
2813         } else {
2814                 (*itm).print(c);
2815         }
2816         its++;
2817         itm++;
2818         for (; itm != m.end(); itm++, its++) {
2819                 c.s << ",";
2820                 if (*its < 0) {
2821                         c.s << "\\overline{";
2822                         (*itm).print(c);
2823                         c.s << "}";
2824                 } else {
2825                         (*itm).print(c);
2826                 }
2827         }
2828         c.s << ")";
2829 }
2830
2831
2832 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
2833                                 evalf_func(zeta2_evalf).
2834                                 eval_func(zeta2_eval).
2835                                 derivative_func(zeta2_deriv).
2836                                 print_func<print_latex>(zeta2_print_latex).
2837                                 do_not_evalf_params().
2838                                 overloaded(2));
2839
2840
2841 } // namespace GiNaC
2842