Fixed bug in harmonic polylogs for complex arguments with positive imaginary part.
[ginac.git] / ginac / inifcns_nstdsums.cpp
1 /** @file inifcns_nstdsums.cpp
2  *
3  *  Implementation of some special functions that have a representation as nested sums.
4  *
5  *  The functions are:
6  *    classical polylogarithm              Li(n,x)
7  *    multiple polylogarithm               Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
8  *    nielsen's generalized polylogarithm  S(n,p,x)
9  *    harmonic polylogarithm               H(m,x) or H(lst(m_1,...,m_k),x)
10  *    multiple zeta value                  zeta(m) or zeta(lst(m_1,...,m_k))
11  *    alternating Euler sum                zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
12  *
13  *  Some remarks:
14  *
15  *    - All formulae used can be looked up in the following publications:
16  *      [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
17  *      [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
18  *      [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
19  *      [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
20  *
21  *    - The order of parameters and arguments of Li and zeta is defined according to the nested sums
22  *      representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
23  *      0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
24  *      number --- notation.
25  *
26  *    - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
27  *      the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product
28  *      x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers.
29  *      If you want to have an alternating Euler sum, you have to give the signs of the parameters as a
30  *      second argument s to zeta(m,s) containing 1 and -1.
31  *
32  *    - The calculation of classical polylogarithms is speed up by using Bernoulli numbers and 
33  *      look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
34  *      [Cra] and [BBB] for speed up.
35  *
36  *    - The functions have no series expansion into nested sums. To do this, you have to convert these functions
37  *      into the appropriate objects from the nestedsums library, do the expansion and convert the
38  *      result back.
39  *
40  *    - Numerical testing of this implementation has been performed by doing a comparison of results
41  *      between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
42  *      by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
43  *      comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
44  *      around |x|=1 along with comparisons to corresponding zeta functions.
45  *
46  */
47
48 /*
49  *  GiNaC Copyright (C) 1999-2004 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 int xnsize = 0;
106
107
108 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
109 // With these numbers the polylogs can be calculated as follows:
110 //   Li_p (x)  =  \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with  u = -log(1-x)
111 //   X_0(n) = B_n (Bernoulli numbers)
112 //   X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
113 // The calculation of Xn depends on X0 and X{n-1}.
114 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
115 // This results in a slightly more complicated algorithm for the X_n.
116 // The first index in Xn corresponds to the index of the polylog minus 2.
117 // The second index in Xn corresponds to the index from the actual sum.
118 void fill_Xn(int n)
119 {
120         // rule of thumb. needs to be improved. TODO
121         const int initsize = Digits * 3 / 2;
122
123         if (n>1) {
124                 // calculate X_2 and higher (corresponding to Li_4 and higher)
125                 std::vector<cln::cl_N> buf(initsize);
126                 std::vector<cln::cl_N>::iterator it = buf.begin();
127                 cln::cl_N result;
128                 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
129                 it++;
130                 for (int i=2; i<=initsize; i++) {
131                         if (i&1) {
132                                 result = 0; // k == 0
133                         } else {
134                                 result = Xn[0][i/2-1]; // k == 0
135                         }
136                         for (int k=1; k<i-1; k++) {
137                                 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
138                                         result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
139                                 }
140                         }
141                         result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
142                         result = result + Xn[n-1][i-1] / (i+1); // k == i
143                         
144                         *it = result;
145                         it++;
146                 }
147                 Xn.push_back(buf);
148         } else if (n==1) {
149                 // special case to handle the X_0 correct
150                 std::vector<cln::cl_N> buf(initsize);
151                 std::vector<cln::cl_N>::iterator it = buf.begin();
152                 cln::cl_N result;
153                 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
154                 it++;
155                 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
156                 it++;
157                 for (int i=3; i<=initsize; i++) {
158                         if (i & 1) {
159                                 result = -Xn[0][(i-3)/2]/2;
160                                 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
161                                 it++;
162                         } else {
163                                 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
164                                 for (int k=1; k<i/2; k++) {
165                                         result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
166                                 }
167                                 *it = result;
168                                 it++;
169                         }
170                 }
171                 Xn.push_back(buf);
172         } else {
173                 // calculate X_0
174                 std::vector<cln::cl_N> buf(initsize/2);
175                 std::vector<cln::cl_N>::iterator it = buf.begin();
176                 for (int i=1; i<=initsize/2; i++) {
177                         *it = bernoulli(i*2).to_cl_N();
178                         it++;
179                 }
180                 Xn.push_back(buf);
181         }
182
183         xnsize++;
184 }
185
186
187 // calculates Li(2,x) without Xn
188 cln::cl_N Li2_do_sum(const cln::cl_N& x)
189 {
190         cln::cl_N res = x;
191         cln::cl_N resbuf;
192         cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
193         cln::cl_I den = 1; // n^2 = 1
194         unsigned i = 3;
195         do {
196                 resbuf = res;
197                 num = num * x;
198                 den = den + i;  // n^2 = 4, 9, 16, ...
199                 i += 2;
200                 res = res + num / den;
201         } while (res != resbuf);
202         return res;
203 }
204
205
206 // calculates Li(2,x) with Xn
207 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
208 {
209         std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
210         cln::cl_N u = -cln::log(1-x);
211         cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
212         cln::cl_N res = u - u*u/4;
213         cln::cl_N resbuf;
214         unsigned i = 1;
215         do {
216                 resbuf = res;
217                 factor = factor * u*u / (2*i * (2*i+1));
218                 res = res + (*it) * factor;
219                 it++; // should we check it? or rely on initsize? ...
220                 i++;
221         } while (res != resbuf);
222         return res;
223 }
224
225
226 // calculates Li(n,x), n>2 without Xn
227 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
228 {
229         cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
230         cln::cl_N res = x;
231         cln::cl_N resbuf;
232         int i=2;
233         do {
234                 resbuf = res;
235                 factor = factor * x;
236                 res = res + factor / cln::expt(cln::cl_I(i),n);
237                 i++;
238         } while (res != resbuf);
239         return res;
240 }
241
242
243 // calculates Li(n,x), n>2 with Xn
244 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
245 {
246         std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
247         cln::cl_N u = -cln::log(1-x);
248         cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
249         cln::cl_N res = u;
250         cln::cl_N resbuf;
251         unsigned i=2;
252         do {
253                 resbuf = res;
254                 factor = factor * u / i;
255                 res = res + (*it) * factor;
256                 it++; // should we check it? or rely on initsize? ...
257                 i++;
258         } while (res != resbuf);
259         return res;
260 }
261
262
263 // forward declaration needed by function Li_projection and C below
264 numeric S_num(int n, int p, const numeric& x);
265
266
267 // helper function for classical polylog Li
268 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
269 {
270         // treat n=2 as special case
271         if (n == 2) {
272                 // check if precalculated X0 exists
273                 if (xnsize == 0) {
274                         fill_Xn(0);
275                 }
276
277                 if (cln::realpart(x) < 0.5) {
278                         // choose the faster algorithm
279                         // the switching point was empirically determined. the optimal point
280                         // depends on hardware, Digits, ... so an approx value is okay.
281                         // it solves also the problem with precision due to the u=-log(1-x) transformation
282                         if (cln::abs(cln::realpart(x)) < 0.25) {
283                                 
284                                 return Li2_do_sum(x);
285                         } else {
286                                 return Li2_do_sum_Xn(x);
287                         }
288                 } else {
289                         // choose the faster algorithm
290                         if (cln::abs(cln::realpart(x)) > 0.75) {
291                                 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
292                         } else {
293                                 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
294                         }
295                 }
296         } else {
297                 // check if precalculated Xn exist
298                 if (n > xnsize+1) {
299                         for (int i=xnsize; i<n-1; i++) {
300                                 fill_Xn(i);
301                         }
302                 }
303
304                 if (cln::realpart(x) < 0.5) {
305                         // choose the faster algorithm
306                         // with n>=12 the "normal" summation always wins against the method with Xn
307                         if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
308                                 return Lin_do_sum(n, x);
309                         } else {
310                                 return Lin_do_sum_Xn(n, x);
311                         }
312                 } else {
313                         cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
314                         for (int j=0; j<n-1; j++) {
315                                 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
316                                                   * cln::expt(cln::log(x), j) / cln::factorial(j);
317                         }
318                         return result;
319                 }
320         }
321 }
322
323
324 // helper function for classical polylog Li
325 numeric Li_num(int n, const numeric& x)
326 {
327         if (n == 1) {
328                 // just a log
329                 return -cln::log(1-x.to_cl_N());
330         }
331         if (x.is_zero()) {
332                 return 0;
333         }
334         if (x == 1) {
335                 // [Kol] (2.22)
336                 return cln::zeta(n);
337         }
338         else if (x == -1) {
339                 // [Kol] (2.22)
340                 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
341         }
342         
343         // what is the desired float format?
344         // first guess: default format
345         cln::float_format_t prec = cln::default_float_format;
346         const cln::cl_N value = x.to_cl_N();
347         // second guess: the argument's format
348         if (!x.real().is_rational())
349                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
350         else if (!x.imag().is_rational())
351                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
352         
353         // [Kol] (5.15)
354         if (cln::abs(value) > 1) {
355                 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
356                 // check if argument is complex. if it is real, the new polylog has to be conjugated.
357                 if (cln::zerop(cln::imagpart(value))) {
358                         if (n & 1) {
359                                 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
360                         }
361                         else {
362                                 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
363                         }
364                 }
365                 else {
366                         if (n & 1) {
367                                 result = result + Li_projection(n, cln::recip(value), prec);
368                         }
369                         else {
370                                 result = result - Li_projection(n, cln::recip(value), prec);
371                         }
372                 }
373                 cln::cl_N add;
374                 for (int j=0; j<n-1; j++) {
375                         add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
376                                     * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
377                 }
378                 result = result - add;
379                 return result;
380         }
381         else {
382                 return Li_projection(n, value, prec);
383         }
384 }
385
386
387 } // end of anonymous namespace
388
389
390 //////////////////////////////////////////////////////////////////////
391 //
392 // Multiple polylogarithm  Li(n,x)
393 //
394 // helper function
395 //
396 //////////////////////////////////////////////////////////////////////
397
398
399 // anonymous namespace for helper function
400 namespace {
401
402
403 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
404 {
405         const int j = s.size();
406
407         std::vector<cln::cl_N> t(j);
408         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
409
410         cln::cl_N t0buf;
411         int q = 0;
412         do {
413                 t0buf = t[0];
414                 // do it once ...
415                 q++;
416                 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
417                 for (int k=j-2; k>=0; k--) {
418                         t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
419                 }
420                 // ... and do it again (to avoid premature drop out due to special arguments)
421                 q++;
422                 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
423                 for (int k=j-2; k>=0; k--) {
424                         t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
425                 }
426         } while (t[0] != t0buf);
427
428         return t[0];
429 }
430
431 // forward declaration for Li_eval()
432 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
433
434
435 } // end of anonymous namespace
436
437
438 //////////////////////////////////////////////////////////////////////
439 //
440 // Classical polylogarithm and multiple polylogarithm  Li(n,x)
441 //
442 // GiNaC function
443 //
444 //////////////////////////////////////////////////////////////////////
445
446
447 static ex Li_evalf(const ex& x1, const ex& x2)
448 {
449         // classical polylogs
450         if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
451                 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
452         }
453         // multiple polylogs
454         else if (is_a<lst>(x1) && is_a<lst>(x2)) {
455                 ex conv = 1;
456                 for (int i=0; i<x1.nops(); i++) {
457                         if (!x1.op(i).info(info_flags::posint)) {
458                                 return Li(x1, x2).hold();
459                         }
460                         if (!is_a<numeric>(x2.op(i))) {
461                                 return Li(x1, x2).hold();
462                         }
463                         conv *= x2.op(i);
464                         if (abs(conv) >= 1) {
465                                 return Li(x1, x2).hold();
466                         }
467                 }
468
469                 std::vector<int> m;
470                 std::vector<cln::cl_N> x;
471                 for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
472                         m.push_back(ex_to<numeric>(x1.op(i)).to_int());
473                         x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
474                 }
475
476                 return numeric(multipleLi_do_sum(m, x));
477         }
478
479         return Li(x1,x2).hold();
480 }
481
482
483 static ex Li_eval(const ex& m_, const ex& x_)
484 {
485         if (m_.nops() < 2) {
486                 ex m;
487                 if (is_a<lst>(m_)) {
488                         m = m_.op(0);
489                 } else {
490                         m = m_;
491                 }
492                 ex x;
493                 if (is_a<lst>(x_)) {
494                         x = x_.op(0);
495                 } else {
496                         x = x_;
497                 }
498                 if (x == _ex0) {
499                         return _ex0;
500                 }
501                 if (x == _ex1) {
502                         return zeta(m);
503                 }
504                 if (x == _ex_1) {
505                         return (pow(2,1-m)-1) * zeta(m);
506                 }
507                 if (m == _ex1) {
508                         return -log(1-x);
509                 }
510                 if (m.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
511                         return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
512                 }
513         } else {
514                 bool ish = true;
515                 bool iszeta = true;
516                 bool iszero = false;
517                 bool doevalf = false;
518                 bool doevalfveto = true;
519                 const lst& m = ex_to<lst>(m_);
520                 const lst& x = ex_to<lst>(x_);
521                 lst::const_iterator itm = m.begin();
522                 lst::const_iterator itx = x.begin();
523                 for (; itm != m.end(); itm++, itx++) {
524                         if (!(*itm).info(info_flags::posint)) {
525                                 return Li(m_, x_).hold();
526                         }
527                         if ((*itx != _ex1) && (*itx != _ex_1)) {
528                                 if (itx != x.begin()) {
529                                         ish = false;
530                                 }
531                                 iszeta = false;
532                         }
533                         if (*itx == _ex0) {
534                                 iszero = true;
535                         }
536                         if (!(*itx).info(info_flags::numeric)) {
537                                 doevalfveto = false;
538                         }
539                         if (!(*itx).info(info_flags::crational)) {
540                                 doevalf = true;
541                         }
542                 }
543                 if (iszeta) {
544                         return zeta(m_, x_);
545                 }
546                 if (iszero) {
547                         return _ex0;
548                 }
549                 if (ish) {
550                         ex pf;
551                         lst newm = convert_parameter_Li_to_H(m, x, pf);
552                         return pf * H(newm, x[0]);
553                 }
554                 if (doevalfveto && doevalf) {
555                         return Li(m_, x_).evalf();
556                 }
557         }
558         return Li(m_, x_).hold();
559 }
560
561
562 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
563 {
564         epvector seq;
565         seq.push_back(expair(Li(m, x), 0));
566         return pseries(rel, seq);
567 }
568
569
570 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
571 {
572         GINAC_ASSERT(deriv_param < 2);
573         if (deriv_param == 0) {
574                 return _ex0;
575         }
576         if (m_.nops() > 1) {
577                 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
578         }
579         ex m;
580         if (is_a<lst>(m_)) {
581                 m = m_.op(0);
582         } else {
583                 m = m_;
584         }
585         ex x;
586         if (is_a<lst>(x_)) {
587                 x = x_.op(0);
588         } else {
589                 x = x_;
590         }
591         if (m > 0) {
592                 return Li(m-1, x) / x;
593         } else {
594                 return 1/(1-x);
595         }
596 }
597
598
599 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
600 {
601         lst m;
602         if (is_a<lst>(m_)) {
603                 m = ex_to<lst>(m_);
604         } else {
605                 m = lst(m_);
606         }
607         lst x;
608         if (is_a<lst>(x_)) {
609                 x = ex_to<lst>(x_);
610         } else {
611                 x = lst(x_);
612         }
613         c.s << "\\mbox{Li}_{";
614         lst::const_iterator itm = m.begin();
615         (*itm).print(c);
616         itm++;
617         for (; itm != m.end(); itm++) {
618                 c.s << ",";
619                 (*itm).print(c);
620         }
621         c.s << "}(";
622         lst::const_iterator itx = x.begin();
623         (*itx).print(c);
624         itx++;
625         for (; itx != x.end(); itx++) {
626                 c.s << ",";
627                 (*itx).print(c);
628         }
629         c.s << ")";
630 }
631
632
633 REGISTER_FUNCTION(Li,
634                   evalf_func(Li_evalf).
635                   eval_func(Li_eval).
636                   series_func(Li_series).
637                   derivative_func(Li_deriv).
638                   print_func<print_latex>(Li_print_latex).
639                   do_not_evalf_params());
640
641
642 //////////////////////////////////////////////////////////////////////
643 //
644 // Nielsen's generalized polylogarithm  S(n,p,x)
645 //
646 // helper functions
647 //
648 //////////////////////////////////////////////////////////////////////
649
650
651 // anonymous namespace for helper functions
652 namespace {
653
654
655 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
656 // see fill_Yn()
657 std::vector<std::vector<cln::cl_N> > Yn;
658 int ynsize = 0; // number of Yn[]
659 int ynlength = 100; // initial length of all Yn[i]
660
661
662 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
663 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
664 // representing S_{n,p}(x).
665 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
666 // equivalent Z-sum.
667 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
668 // representing S_{n,p}(x).
669 // The calculation of Y_n uses the values from Y_{n-1}.
670 void fill_Yn(int n, const cln::float_format_t& prec)
671 {
672         const int initsize = ynlength;
673         //const int initsize = initsize_Yn;
674         cln::cl_N one = cln::cl_float(1, prec);
675
676         if (n) {
677                 std::vector<cln::cl_N> buf(initsize);
678                 std::vector<cln::cl_N>::iterator it = buf.begin();
679                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
680                 *it = (*itprev) / cln::cl_N(n+1) * one;
681                 it++;
682                 itprev++;
683                 // sums with an index smaller than the depth are zero and need not to be calculated.
684                 // calculation starts with depth, which is n+2)
685                 for (int i=n+2; i<=initsize+n; i++) {
686                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
687                         it++;
688                         itprev++;
689                 }
690                 Yn.push_back(buf);
691         } else {
692                 std::vector<cln::cl_N> buf(initsize);
693                 std::vector<cln::cl_N>::iterator it = buf.begin();
694                 *it = 1 * one;
695                 it++;
696                 for (int i=2; i<=initsize; i++) {
697                         *it = *(it-1) + 1 / cln::cl_N(i) * one;
698                         it++;
699                 }
700                 Yn.push_back(buf);
701         }
702         ynsize++;
703 }
704
705
706 // make Yn longer ... 
707 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
708 {
709
710         cln::cl_N one = cln::cl_float(1, prec);
711
712         Yn[0].resize(newsize);
713         std::vector<cln::cl_N>::iterator it = Yn[0].begin();
714         it += ynlength;
715         for (int i=ynlength+1; i<=newsize; i++) {
716                 *it = *(it-1) + 1 / cln::cl_N(i) * one;
717                 it++;
718         }
719
720         for (int n=1; n<ynsize; n++) {
721                 Yn[n].resize(newsize);
722                 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
723                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
724                 it += ynlength;
725                 itprev += ynlength;
726                 for (int i=ynlength+n+1; i<=newsize+n; i++) {
727                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
728                         it++;
729                         itprev++;
730                 }
731         }
732         
733         ynlength = newsize;
734 }
735
736
737 // helper function for S(n,p,x)
738 // [Kol] (7.2)
739 cln::cl_N C(int n, int p)
740 {
741         cln::cl_N result;
742
743         for (int k=0; k<p; k++) {
744                 for (int j=0; j<=(n+k-1)/2; j++) {
745                         if (k == 0) {
746                                 if (n & 1) {
747                                         if (j & 1) {
748                                                 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
749                                         }
750                                         else {
751                                                 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
752                                         }
753                                 }
754                         }
755                         else {
756                                 if (k & 1) {
757                                         if (j & 1) {
758                                                 result = result + cln::factorial(n+k-1)
759                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
760                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
761                                         }
762                                         else {
763                                                 result = result - cln::factorial(n+k-1)
764                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
765                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
766                                         }
767                                 }
768                                 else {
769                                         if (j & 1) {
770                                                 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
771                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
772                                         }
773                                         else {
774                                                 result = result + cln::factorial(n+k-1)
775                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
776                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
777                                         }
778                                 }
779                         }
780                 }
781         }
782         int np = n+p;
783         if ((np-1) & 1) {
784                 if (((np)/2+n) & 1) {
785                         result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
786                 }
787                 else {
788                         result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
789                 }
790         }
791
792         return result;
793 }
794
795
796 // helper function for S(n,p,x)
797 // [Kol] remark to (9.1)
798 cln::cl_N a_k(int k)
799 {
800         cln::cl_N result;
801
802         if (k == 0) {
803                 return 1;
804         }
805
806         result = result;
807         for (int m=2; m<=k; m++) {
808                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
809         }
810
811         return -result / k;
812 }
813
814
815 // helper function for S(n,p,x)
816 // [Kol] remark to (9.1)
817 cln::cl_N b_k(int k)
818 {
819         cln::cl_N result;
820
821         if (k == 0) {
822                 return 1;
823         }
824
825         result = result;
826         for (int m=2; m<=k; m++) {
827                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
828         }
829
830         return result / k;
831 }
832
833
834 // helper function for S(n,p,x)
835 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
836 {
837         if (p==1) {
838                 return Li_projection(n+1, x, prec);
839         }
840         
841         // check if precalculated values are sufficient
842         if (p > ynsize+1) {
843                 for (int i=ynsize; i<p-1; i++) {
844                         fill_Yn(i, prec);
845                 }
846         }
847
848         // should be done otherwise
849         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
850         cln::cl_N xf = x * one;
851         //cln::cl_N xf = x * cln::cl_float(1, prec);
852
853         cln::cl_N res;
854         cln::cl_N resbuf;
855         cln::cl_N factor = cln::expt(xf, p);
856         int i = p;
857         do {
858                 resbuf = res;
859                 if (i-p >= ynlength) {
860                         // make Yn longer
861                         make_Yn_longer(ynlength*2, prec);
862                 }
863                 res = res + factor / cln::expt(cln::cl_I(i),n+1) * Yn[p-2][i-p]; // should we check it? or rely on magic number? ...
864                 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
865                 factor = factor * xf;
866                 i++;
867         } while (res != resbuf);
868         
869         return res;
870 }
871
872
873 // helper function for S(n,p,x)
874 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
875 {
876         // [Kol] (5.3)
877         if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
878
879                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
880                                    * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
881
882                 for (int s=0; s<n; s++) {
883                         cln::cl_N res2;
884                         for (int r=0; r<p; r++) {
885                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
886                                               * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
887                         }
888                         result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
889                 }
890
891                 return result;
892         }
893         
894         return S_do_sum(n, p, x, prec);
895 }
896
897
898 // helper function for S(n,p,x)
899 numeric S_num(int n, int p, const numeric& x)
900 {
901         if (x == 1) {
902                 if (n == 1) {
903                     // [Kol] (2.22) with (2.21)
904                         return cln::zeta(p+1);
905                 }
906
907                 if (p == 1) {
908                     // [Kol] (2.22)
909                         return cln::zeta(n+1);
910                 }
911
912                 // [Kol] (9.1)
913                 cln::cl_N result;
914                 for (int nu=0; nu<n; nu++) {
915                         for (int rho=0; rho<=p; rho++) {
916                                 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
917                                                   * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
918                         }
919                 }
920                 result = result * cln::expt(cln::cl_I(-1),n+p-1);
921
922                 return result;
923         }
924         else if (x == -1) {
925                 // [Kol] (2.22)
926                 if (p == 1) {
927                         return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
928                 }
929 //              throw std::runtime_error("don't know how to evaluate this function!");
930         }
931
932         // what is the desired float format?
933         // first guess: default format
934         cln::float_format_t prec = cln::default_float_format;
935         const cln::cl_N value = x.to_cl_N();
936         // second guess: the argument's format
937         if (!x.real().is_rational())
938                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
939         else if (!x.imag().is_rational())
940                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
941
942
943         // [Kol] (5.3)
944         if (cln::realpart(value) < -0.5) {
945
946                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
947                                    * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
948
949                 for (int s=0; s<n; s++) {
950                         cln::cl_N res2;
951                         for (int r=0; r<p; r++) {
952                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
953                                               * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
954                         }
955                         result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
956                 }
957
958                 return result;
959                 
960         }
961         // [Kol] (5.12)
962         if (cln::abs(value) > 1) {
963                 
964                 cln::cl_N result;
965
966                 for (int s=0; s<p; s++) {
967                         for (int r=0; r<=s; r++) {
968                                 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
969                                                   / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
970                                                   * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
971                         }
972                 }
973                 result = result * cln::expt(cln::cl_I(-1),n);
974
975                 cln::cl_N res2;
976                 for (int r=0; r<n; r++) {
977                         res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
978                 }
979                 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
980
981                 result = result + cln::expt(cln::cl_I(-1),p) * res2;
982
983                 return result;
984         }
985         else {
986                 return S_projection(n, p, value, prec);
987         }
988 }
989
990
991 } // end of anonymous namespace
992
993
994 //////////////////////////////////////////////////////////////////////
995 //
996 // Nielsen's generalized polylogarithm  S(n,p,x)
997 //
998 // GiNaC function
999 //
1000 //////////////////////////////////////////////////////////////////////
1001
1002
1003 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1004 {
1005         if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a<numeric>(x)) {
1006                 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1007         }
1008         return S(n, p, x).hold();
1009 }
1010
1011
1012 static ex S_eval(const ex& n, const ex& p, const ex& x)
1013 {
1014         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1015                 if (x == 0) {
1016                         return _ex0;
1017                 }
1018                 if (x == 1) {
1019                         lst m(n+1);
1020                         for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
1021                                 m.append(1);
1022                         }
1023                         return zeta(m);
1024                 }
1025                 if (p == 1) {
1026                         return Li(n+1, x);
1027                 }
1028                 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
1029                         return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1030                 }
1031         }
1032         return S(n, p, x).hold();
1033 }
1034
1035
1036 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
1037 {
1038         epvector seq;
1039         seq.push_back(expair(S(n, p, x), 0));
1040         return pseries(rel, seq);
1041 }
1042
1043
1044 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
1045 {
1046         GINAC_ASSERT(deriv_param < 3);
1047         if (deriv_param < 2) {
1048                 return _ex0;
1049         }
1050         if (n > 0) {
1051                 return S(n-1, p, x) / x;
1052         } else {
1053                 return S(n, p-1, x) / (1-x);
1054         }
1055 }
1056
1057
1058 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
1059 {
1060         c.s << "\\mbox{S}_{";
1061         n.print(c);
1062         c.s << ",";
1063         p.print(c);
1064         c.s << "}(";
1065         x.print(c);
1066         c.s << ")";
1067 }
1068
1069
1070 REGISTER_FUNCTION(S,
1071                   evalf_func(S_evalf).
1072                   eval_func(S_eval).
1073                   series_func(S_series).
1074                   derivative_func(S_deriv).
1075                   print_func<print_latex>(S_print_latex).
1076                   do_not_evalf_params());
1077
1078
1079 //////////////////////////////////////////////////////////////////////
1080 //
1081 // Harmonic polylogarithm  H(m,x)
1082 //
1083 // helper functions
1084 //
1085 //////////////////////////////////////////////////////////////////////
1086
1087
1088 // anonymous namespace for helper functions
1089 namespace {
1090
1091         
1092 // regulates the pole (used by 1/x-transformation)
1093 symbol H_polesign("IMSIGN");
1094
1095
1096 // convert parameters from H to Li representation
1097 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
1098 // returns true if some parameters are negative
1099 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
1100 {
1101         // expand parameter list
1102         lst mexp;
1103         for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
1104                 if (*it > 1) {
1105                         for (ex count=*it-1; count > 0; count--) {
1106                                 mexp.append(0);
1107                         }
1108                         mexp.append(1);
1109                 } else if (*it < -1) {
1110                         for (ex count=*it+1; count < 0; count++) {
1111                                 mexp.append(0);
1112                         }
1113                         mexp.append(-1);
1114                 } else {
1115                         mexp.append(*it);
1116                 }
1117         }
1118         
1119         ex signum = 1;
1120         pf = 1;
1121         bool has_negative_parameters = false;
1122         ex acc = 1;
1123         for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
1124                 if (*it == 0) {
1125                         acc++;
1126                         continue;
1127                 }
1128                 if (*it > 0) {
1129                         m.append((*it+acc-1) * signum);
1130                 } else {
1131                         m.append((*it-acc+1) * signum);
1132                 }
1133                 acc = 1;
1134                 signum = *it;
1135                 pf *= *it;
1136                 if (pf < 0) {
1137                         has_negative_parameters = true;
1138                 }
1139         }
1140         if (has_negative_parameters) {
1141                 for (int i=0; i<m.nops(); i++) {
1142                         if (m.op(i) < 0) {
1143                                 m.let_op(i) = -m.op(i);
1144                                 s.append(-1);
1145                         } else {
1146                                 s.append(1);
1147                         }
1148                 }
1149         }
1150         
1151         return has_negative_parameters;
1152 }
1153
1154
1155 // recursivly transforms H to corresponding multiple polylogarithms
1156 struct map_trafo_H_convert_to_Li : public map_function
1157 {
1158         ex operator()(const ex& e)
1159         {
1160                 if (is_a<add>(e) || is_a<mul>(e)) {
1161                         return e.map(*this);
1162                 }
1163                 if (is_a<function>(e)) {
1164                         std::string name = ex_to<function>(e).get_name();
1165                         if (name == "H") {
1166                                 lst parameter;
1167                                 if (is_a<lst>(e.op(0))) {
1168                                                 parameter = ex_to<lst>(e.op(0));
1169                                 } else {
1170                                         parameter = lst(e.op(0));
1171                                 }
1172                                 ex arg = e.op(1);
1173
1174                                 lst m;
1175                                 lst s;
1176                                 ex pf;
1177                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1178                                         s.let_op(0) = s.op(0) * arg;
1179                                         return pf * Li(m, s).hold();
1180                                 } else {
1181                                         for (int i=0; i<m.nops(); i++) {
1182                                                 s.append(1);
1183                                         }
1184                                         s.let_op(0) = s.op(0) * arg;
1185                                         return Li(m, s).hold();
1186                                 }
1187                         }
1188                 }
1189                 return e;
1190         }
1191 };
1192
1193
1194 // recursivly transforms H to corresponding zetas
1195 struct map_trafo_H_convert_to_zeta : public map_function
1196 {
1197         ex operator()(const ex& e)
1198         {
1199                 if (is_a<add>(e) || is_a<mul>(e)) {
1200                         return e.map(*this);
1201                 }
1202                 if (is_a<function>(e)) {
1203                         std::string name = ex_to<function>(e).get_name();
1204                         if (name == "H") {
1205                                 lst parameter;
1206                                 if (is_a<lst>(e.op(0))) {
1207                                                 parameter = ex_to<lst>(e.op(0));
1208                                 } else {
1209                                         parameter = lst(e.op(0));
1210                                 }
1211
1212                                 lst m;
1213                                 lst s;
1214                                 ex pf;
1215                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1216                                         return pf * zeta(m, s);
1217                                 } else {
1218                                         return zeta(m);
1219                                 }
1220                         }
1221                 }
1222                 return e;
1223         }
1224 };
1225
1226
1227 // remove trailing zeros from H-parameters
1228 struct map_trafo_H_reduce_trailing_zeros : public map_function
1229 {
1230         ex operator()(const ex& e)
1231         {
1232                 if (is_a<add>(e) || is_a<mul>(e)) {
1233                         return e.map(*this);
1234                 }
1235                 if (is_a<function>(e)) {
1236                         std::string name = ex_to<function>(e).get_name();
1237                         if (name == "H") {
1238                                 lst parameter;
1239                                 if (is_a<lst>(e.op(0))) {
1240                                         parameter = ex_to<lst>(e.op(0));
1241                                 } else {
1242                                         parameter = lst(e.op(0));
1243                                 }
1244                                 ex arg = e.op(1);
1245                                 if (parameter.op(parameter.nops()-1) == 0) {
1246                                         
1247                                         //
1248                                         if (parameter.nops() == 1) {
1249                                                 return log(arg);
1250                                         }
1251                                         
1252                                         //
1253                                         lst::const_iterator it = parameter.begin();
1254                                         while ((it != parameter.end()) && (*it == 0)) {
1255                                                 it++;
1256                                         }
1257                                         if (it == parameter.end()) {
1258                                                 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1259                                         }
1260                                         
1261                                         //
1262                                         parameter.remove_last();
1263                                         int lastentry = parameter.nops();
1264                                         while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1265                                                 lastentry--;
1266                                         }
1267                                         
1268                                         //
1269                                         ex result = log(arg) * H(parameter,arg).hold();
1270                                         ex acc = 0;
1271                                         for (ex i=0; i<lastentry; i++) {
1272                                                 if (parameter[i] > 0) {
1273                                                         parameter[i]++;
1274                                                         result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
1275                                                         parameter[i]--;
1276                                                         acc = 0;
1277                                                 } else if (parameter[i] < 0) {
1278                                                         parameter[i]--;
1279                                                         result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
1280                                                         parameter[i]++;
1281                                                         acc = 0;
1282                                                 } else {
1283                                                         acc++;
1284                                                 }
1285                                         }
1286                                         
1287                                         if (lastentry < parameter.nops()) {
1288                                                 result = result / (parameter.nops()-lastentry+1);
1289                                                 return result.map(*this);
1290                                         } else {
1291                                                 return result;
1292                                         }
1293                                 }
1294                         }
1295                 }
1296                 return e;
1297         }
1298 };
1299
1300
1301 // returns an expression with zeta functions corresponding to the parameter list for H
1302 ex convert_H_to_zeta(const lst& m)
1303 {
1304         symbol xtemp("xtemp");
1305         map_trafo_H_reduce_trailing_zeros filter;
1306         map_trafo_H_convert_to_zeta filter2;
1307         return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
1308 }
1309
1310
1311 // convert signs form Li to H representation
1312 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
1313 {
1314         lst res;
1315         lst::const_iterator itm = m.begin();
1316         lst::const_iterator itx = ++x.begin();
1317         ex signum = _ex1;
1318         pf = _ex1;
1319         res.append(*itm);
1320         itm++;
1321         while (itx != x.end()) {
1322                 signum *= *itx;
1323                 pf *= signum;
1324                 res.append((*itm) * signum);
1325                 itm++;
1326                 itx++;
1327         }
1328         return res;
1329 }
1330
1331
1332 // multiplies an one-dimensional H with another H
1333 // [ReV] (18)
1334 ex trafo_H_mult(const ex& h1, const ex& h2)
1335 {
1336         ex res;
1337         ex hshort;
1338         lst hlong;
1339         ex h1nops = h1.op(0).nops();
1340         ex h2nops = h2.op(0).nops();
1341         if (h1nops > 1) {
1342                 hshort = h2.op(0).op(0);
1343                 hlong = ex_to<lst>(h1.op(0));
1344         } else {
1345                 hshort = h1.op(0).op(0);
1346                 if (h2nops > 1) {
1347                         hlong = ex_to<lst>(h2.op(0));
1348                 } else {
1349                         hlong = h2.op(0).op(0);
1350                 }
1351         }
1352         for (int i=0; i<=hlong.nops(); i++) {
1353                 lst newparameter;
1354                 int j=0;
1355                 for (; j<i; j++) {
1356                         newparameter.append(hlong[j]);
1357                 }
1358                 newparameter.append(hshort);
1359                 for (; j<hlong.nops(); j++) {
1360                         newparameter.append(hlong[j]);
1361                 }
1362                 res += H(newparameter, h1.op(1)).hold();
1363         }
1364         return res;
1365 }
1366
1367
1368 // applies trafo_H_mult recursively on expressions
1369 struct map_trafo_H_mult : public map_function
1370 {
1371         ex operator()(const ex& e)
1372         {
1373                 if (is_a<add>(e)) {
1374                         return e.map(*this);
1375                 }
1376
1377                 if (is_a<mul>(e)) {
1378
1379                         ex result = 1;
1380                         ex firstH;
1381                         lst Hlst;
1382                         for (int pos=0; pos<e.nops(); pos++) {
1383                                 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1384                                         std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1385                                         if (name == "H") {
1386                                                 for (ex i=0; i<e.op(pos).op(1); i++) {
1387                                                         Hlst.append(e.op(pos).op(0));
1388                                                 }
1389                                                 continue;
1390                                         }
1391                                 } else if (is_a<function>(e.op(pos))) {
1392                                         std::string name = ex_to<function>(e.op(pos)).get_name();
1393                                         if (name == "H") {
1394                                                 if (e.op(pos).op(0).nops() > 1) {
1395                                                         firstH = e.op(pos);
1396                                                 } else {
1397                                                         Hlst.append(e.op(pos));
1398                                                 }
1399                                                 continue;
1400                                         }
1401                                 }
1402                                 result *= e.op(pos);
1403                         }
1404                         if (firstH == 0) {
1405                                 if (Hlst.nops() > 0) {
1406                                         firstH = Hlst[Hlst.nops()-1];
1407                                         Hlst.remove_last();
1408                                 } else {
1409                                         return e;
1410                                 }
1411                         }
1412
1413                         if (Hlst.nops() > 0) {
1414                                 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1415                                 result *= buffer;
1416                                 for (int i=1; i<Hlst.nops(); i++) {
1417                                         result *= Hlst.op(i);
1418                                 }
1419                                 result = result.expand();
1420                                 map_trafo_H_mult recursion;
1421                                 return recursion(result);
1422                         } else {
1423                                 return e;
1424                         }
1425
1426                 }
1427                 return e;
1428         }
1429 };
1430
1431
1432 // do integration [ReV] (55)
1433 // put parameter 0 in front of existing parameters
1434 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
1435 {
1436         ex h;
1437         std::string name;
1438         if (is_a<function>(e)) {
1439                 name = ex_to<function>(e).get_name();
1440         }
1441         if (name == "H") {
1442                 h = e;
1443         } else {
1444                 for (int i=0; i<e.nops(); i++) {
1445                         if (is_a<function>(e.op(i))) {
1446                                 std::string name = ex_to<function>(e.op(i)).get_name();
1447                                 if (name == "H") {
1448                                         h = e.op(i);
1449                                 }
1450                         }
1451                 }
1452         }
1453         if (h != 0) {
1454                 lst newparameter = ex_to<lst>(h.op(0));
1455                 newparameter.prepend(0);
1456                 ex addzeta = convert_H_to_zeta(newparameter);
1457                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1458         } else {
1459                 return e * (-H(lst(0),1/arg).hold());
1460         }
1461 }
1462
1463
1464 // do integration [ReV] (55)
1465 // put parameter -1 in front of existing parameters
1466 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
1467 {
1468         ex h;
1469         std::string name;
1470         if (is_a<function>(e)) {
1471                 name = ex_to<function>(e).get_name();
1472         }
1473         if (name == "H") {
1474                 h = e;
1475         } else {
1476                 for (int i=0; i<e.nops(); i++) {
1477                         if (is_a<function>(e.op(i))) {
1478                                 std::string name = ex_to<function>(e.op(i)).get_name();
1479                                 if (name == "H") {
1480                                         h = e.op(i);
1481                                 }
1482                         }
1483                 }
1484         }
1485         if (h != 0) {
1486                 lst newparameter = ex_to<lst>(h.op(0));
1487                 newparameter.prepend(-1);
1488                 ex addzeta = convert_H_to_zeta(newparameter);
1489                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1490         } else {
1491                 ex addzeta = convert_H_to_zeta(lst(-1));
1492                 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
1493         }
1494 }
1495
1496
1497 // do integration [ReV] (55)
1498 // put parameter -1 in front of existing parameters
1499 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
1500 {
1501         ex h;
1502         std::string name;
1503         if (is_a<function>(e)) {
1504                 name = ex_to<function>(e).get_name();
1505         }
1506         if (name == "H") {
1507                 h = e;
1508         } else {
1509                 for (int i=0; i<e.nops(); i++) {
1510                         if (is_a<function>(e.op(i))) {
1511                                 std::string name = ex_to<function>(e.op(i)).get_name();
1512                                 if (name == "H") {
1513                                         h = e.op(i);
1514                                 }
1515                         }
1516                 }
1517         }
1518         if (h != 0) {
1519                 lst newparameter = ex_to<lst>(h.op(0));
1520                 newparameter.prepend(-1);
1521                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1522         } else {
1523                 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
1524         }
1525 }
1526
1527
1528 // do integration [ReV] (55)
1529 // put parameter 1 in front of existing parameters
1530 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
1531 {
1532         ex h;
1533         std::string name;
1534         if (is_a<function>(e)) {
1535                 name = ex_to<function>(e).get_name();
1536         }
1537         if (name == "H") {
1538                 h = e;
1539         } else {
1540                 for (int i=0; i<e.nops(); i++) {
1541                         if (is_a<function>(e.op(i))) {
1542                                 std::string name = ex_to<function>(e.op(i)).get_name();
1543                                 if (name == "H") {
1544                                         h = e.op(i);
1545                                 }
1546                         }
1547                 }
1548         }
1549         if (h != 0) {
1550                 lst newparameter = ex_to<lst>(h.op(0));
1551                 newparameter.prepend(1);
1552                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1553         } else {
1554                 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
1555         }
1556 }
1557
1558
1559 // do x -> 1/x transformation
1560 struct map_trafo_H_1overx : public map_function
1561 {
1562         ex operator()(const ex& e)
1563         {
1564                 if (is_a<add>(e) || is_a<mul>(e)) {
1565                         return e.map(*this);
1566                 }
1567
1568                 if (is_a<function>(e)) {
1569                         std::string name = ex_to<function>(e).get_name();
1570                         if (name == "H") {
1571
1572                                 lst parameter = ex_to<lst>(e.op(0));
1573                                 ex arg = e.op(1);
1574
1575                                 // special cases if all parameters are either 0, 1 or -1
1576                                 bool allthesame = true;
1577                                 if (parameter.op(0) == 0) {
1578                                         for (int i=1; i<parameter.nops(); i++) {
1579                                                 if (parameter.op(i) != 0) {
1580                                                         allthesame = false;
1581                                                         break;
1582                                                 }
1583                                         }
1584                                         if (allthesame) {
1585                                                 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1586                                         }
1587                                 } else if (parameter.op(0) == -1) {
1588                                         for (int i=1; i<parameter.nops(); i++) {
1589                                                 if (parameter.op(i) != -1) {
1590                                                         allthesame = false;
1591                                                         break;
1592                                                 }
1593                                         }
1594                                         if (allthesame) {
1595                                                 map_trafo_H_mult unify;
1596                                                 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
1597                                                        / factorial(parameter.nops())).expand());
1598                                         }
1599                                 } else {
1600                                         for (int i=1; i<parameter.nops(); i++) {
1601                                                 if (parameter.op(i) != 1) {
1602                                                         allthesame = false;
1603                                                         break;
1604                                                 }
1605                                         }
1606                                         if (allthesame) {
1607                                                 map_trafo_H_mult unify;
1608                                                 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
1609                                                        / factorial(parameter.nops())).expand());
1610                                         }
1611                                 }
1612
1613                                 lst newparameter = parameter;
1614                                 newparameter.remove_first();
1615
1616                                 if (parameter.op(0) == 0) {
1617                                         
1618                                         // leading zero
1619                                         ex res = convert_H_to_zeta(parameter);
1620                                         map_trafo_H_1overx recursion;
1621                                         ex buffer = recursion(H(newparameter, arg).hold());
1622                                         if (is_a<add>(buffer)) {
1623                                                 for (int i=0; i<buffer.nops(); i++) {
1624                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
1625                                                 }
1626                                         } else {
1627                                                 res += trafo_H_1tx_prepend_zero(buffer, arg);
1628                                         }
1629                                         return res;
1630
1631                                 } else if (parameter.op(0) == -1) {
1632
1633                                         // leading negative one
1634                                         ex res = convert_H_to_zeta(parameter);
1635                                         map_trafo_H_1overx recursion;
1636                                         ex buffer = recursion(H(newparameter, arg).hold());
1637                                         if (is_a<add>(buffer)) {
1638                                                 for (int i=0; i<buffer.nops(); i++) {
1639                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
1640                                                 }
1641                                         } else {
1642                                                 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
1643                                         }
1644                                         return res;
1645
1646                                 } else {
1647
1648                                         // leading one
1649                                         map_trafo_H_1overx recursion;
1650                                         map_trafo_H_mult unify;
1651                                         ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1652                                         int firstzero = 0;
1653                                         while (parameter.op(firstzero) == 1) {
1654                                                 firstzero++;
1655                                         }
1656                                         for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1657                                                 lst newparameter;
1658                                                 int j=0;
1659                                                 for (; j<=i; j++) {
1660                                                         newparameter.append(parameter[j+1]);
1661                                                 }
1662                                                 newparameter.append(1);
1663                                                 for (; j<parameter.nops()-1; j++) {
1664                                                         newparameter.append(parameter[j+1]);
1665                                                 }
1666                                                 res -= H(newparameter, arg).hold();
1667                                         }
1668                                         res = recursion(res).expand() / firstzero;
1669                                         return unify(res);
1670
1671                                 }
1672
1673                         }
1674                 }
1675                 return e;
1676         }
1677 };
1678
1679
1680 // do x -> (1-x)/(1+x) transformation
1681 struct map_trafo_H_1mxt1px : public map_function
1682 {
1683         ex operator()(const ex& e)
1684         {
1685                 if (is_a<add>(e) || is_a<mul>(e)) {
1686                         return e.map(*this);
1687                 }
1688
1689                 if (is_a<function>(e)) {
1690                         std::string name = ex_to<function>(e).get_name();
1691                         if (name == "H") {
1692
1693                                 lst parameter = ex_to<lst>(e.op(0));
1694                                 ex arg = e.op(1);
1695
1696                                 // special cases if all parameters are either 0, 1 or -1
1697                                 bool allthesame = true;
1698                                 if (parameter.op(0) == 0) {
1699                                         for (int i=1; i<parameter.nops(); i++) {
1700                                                 if (parameter.op(i) != 0) {
1701                                                         allthesame = false;
1702                                                         break;
1703                                                 }
1704                                         }
1705                                         if (allthesame) {
1706                                                 map_trafo_H_mult unify;
1707                                                 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1708                                                        / factorial(parameter.nops())).expand());
1709                                         }
1710                                 } else if (parameter.op(0) == -1) {
1711                                         for (int i=1; i<parameter.nops(); i++) {
1712                                                 if (parameter.op(i) != -1) {
1713                                                         allthesame = false;
1714                                                         break;
1715                                                 }
1716                                         }
1717                                         if (allthesame) {
1718                                                 map_trafo_H_mult unify;
1719                                                 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1720                                                        / factorial(parameter.nops())).expand());
1721                                         }
1722                                 } else {
1723                                         for (int i=1; i<parameter.nops(); i++) {
1724                                                 if (parameter.op(i) != 1) {
1725                                                         allthesame = false;
1726                                                         break;
1727                                                 }
1728                                         }
1729                                         if (allthesame) {
1730                                                 map_trafo_H_mult unify;
1731                                                 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1732                                                        / factorial(parameter.nops())).expand());
1733                                         }
1734                                 }
1735
1736                                 lst newparameter = parameter;
1737                                 newparameter.remove_first();
1738
1739                                 if (parameter.op(0) == 0) {
1740
1741                                         // leading zero
1742                                         ex res = convert_H_to_zeta(parameter);
1743                                         map_trafo_H_1mxt1px recursion;
1744                                         ex buffer = recursion(H(newparameter, arg).hold());
1745                                         if (is_a<add>(buffer)) {
1746                                                 for (int i=0; i<buffer.nops(); i++) {
1747                                                         res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1748                                                 }
1749                                         } else {
1750                                                 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1751                                         }
1752                                         return res;
1753
1754                                 } else if (parameter.op(0) == -1) {
1755
1756                                         // leading negative one
1757                                         ex res = convert_H_to_zeta(parameter);
1758                                         map_trafo_H_1mxt1px recursion;
1759                                         ex buffer = recursion(H(newparameter, arg).hold());
1760                                         if (is_a<add>(buffer)) {
1761                                                 for (int i=0; i<buffer.nops(); i++) {
1762                                                         res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1763                                                 }
1764                                         } else {
1765                                                 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1766                                         }
1767                                         return res;
1768
1769                                 } else {
1770
1771                                         // leading one
1772                                         map_trafo_H_1mxt1px recursion;
1773                                         map_trafo_H_mult unify;
1774                                         ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1775                                         int firstzero = 0;
1776                                         while (parameter.op(firstzero) == 1) {
1777                                                 firstzero++;
1778                                         }
1779                                         for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1780                                                 lst newparameter;
1781                                                 int j=0;
1782                                                 for (; j<=i; j++) {
1783                                                         newparameter.append(parameter[j+1]);
1784                                                 }
1785                                                 newparameter.append(1);
1786                                                 for (; j<parameter.nops()-1; j++) {
1787                                                         newparameter.append(parameter[j+1]);
1788                                                 }
1789                                                 res -= H(newparameter, arg).hold();
1790                                         }
1791                                         res = recursion(res).expand() / firstzero;
1792                                         return unify(res);
1793
1794                                 }
1795
1796                         }
1797                 }
1798                 return e;
1799         }
1800 };
1801
1802
1803 // do the actual summation.
1804 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1805 {
1806         const int j = m.size();
1807
1808         std::vector<cln::cl_N> t(j);
1809
1810         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1811         cln::cl_N factor = cln::expt(x, j) * one;
1812         cln::cl_N t0buf;
1813         int q = 0;
1814         do {
1815                 t0buf = t[0];
1816                 q++;
1817                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1818                 for (int k=j-2; k>=1; k--) {
1819                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1820                 }
1821                 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1822                 factor = factor * x;
1823         } while (t[0] != t0buf);
1824
1825         return t[0];
1826 }
1827
1828
1829 } // end of anonymous namespace
1830
1831
1832 //////////////////////////////////////////////////////////////////////
1833 //
1834 // Harmonic polylogarithm  H(m,x)
1835 //
1836 // GiNaC function
1837 //
1838 //////////////////////////////////////////////////////////////////////
1839
1840
1841 static ex H_evalf(const ex& x1, const ex& x2)
1842 {
1843         if (is_a<lst>(x1) && is_a<numeric>(x2)) {
1844                 for (int i=0; i<x1.nops(); i++) {
1845                         if (!x1.op(i).info(info_flags::integer)) {
1846                                 return H(x1,x2).hold();
1847                         }
1848                 }
1849                 if (x1.nops() < 1) {
1850                         return H(x1,x2).hold();
1851                 }
1852
1853                 cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
1854                 
1855                 const lst& morg = ex_to<lst>(x1);
1856                 // remove trailing zeros ...
1857                 if (*(--morg.end()) == 0) {
1858                         symbol xtemp("xtemp");
1859                         map_trafo_H_reduce_trailing_zeros filter;
1860                         return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
1861                 }
1862                 // ... and expand parameter notation
1863                 lst m;
1864                 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
1865                         if (*it > 1) {
1866                                 for (ex count=*it-1; count > 0; count--) {
1867                                         m.append(0);
1868                                 }
1869                                 m.append(1);
1870                         } else if (*it < -1) {
1871                                 for (ex count=*it+1; count < 0; count++) {
1872                                         m.append(0);
1873                                 }
1874                                 m.append(-1);
1875                         } else {
1876                                 m.append(*it);
1877                         }
1878                 }
1879
1880                 // since the transformations produce a lot of terms, they are only efficient for
1881                 // argument near one.
1882                 // no transformation needed -> do summation
1883                 if (cln::abs(x) < 0.95) {
1884                         lst m_lst;
1885                         lst s_lst;
1886                         ex pf;
1887                         if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
1888                                 // negative parameters -> s_lst is filled
1889                                 std::vector<int> m_int;
1890                                 std::vector<cln::cl_N> x_cln;
1891                                 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin(); 
1892                                      it_int != m_lst.end(); it_int++, it_cln++) {
1893                                         m_int.push_back(ex_to<numeric>(*it_int).to_int());
1894                                         x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
1895                                 }
1896                                 x_cln.front() = x_cln.front() * x;
1897                                 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
1898                         } else {
1899                                 // only positive parameters
1900                                 //TODO
1901                                 if (m_lst.nops() == 1) {
1902                                         return Li(m_lst.op(0), x2).evalf();
1903                                 }
1904                                 std::vector<int> m_int;
1905                                 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
1906                                         m_int.push_back(ex_to<numeric>(*it).to_int());
1907                                 }
1908                                 return numeric(H_do_sum(m_int, x));
1909                         }
1910                 }
1911
1912                 ex res = 1;     
1913                 
1914                 // ensure that the realpart of the argument is positive
1915                 if (cln::realpart(x) < 0) {
1916                         x = -x;
1917                         for (int i=0; i<m.nops(); i++) {
1918                                 if (m.op(i) != 0) {
1919                                         m.let_op(i) = -m.op(i);
1920                                         res *= -1;
1921                                 }
1922                         }
1923                 }
1924
1925                 // choose transformations
1926                 symbol xtemp("xtemp");
1927                 if (cln::abs(x-1) < 1.4142) {
1928                         // x -> (1-x)/(1+x)
1929                         map_trafo_H_1mxt1px trafo;
1930                         res *= trafo(H(m, xtemp));
1931                 } else {
1932                         // x -> 1/x
1933                         map_trafo_H_1overx trafo;
1934                         res *= trafo(H(m, xtemp));
1935                         if (cln::imagpart(x) <= 0) {
1936                                 res = res.subs(H_polesign == -I*Pi);
1937                         } else {
1938                                 res = res.subs(H_polesign == I*Pi);
1939                         }
1940                 }
1941
1942                 // simplify result
1943 // TODO
1944 //              map_trafo_H_convert converter;
1945 //              res = converter(res).expand();
1946 //              lst ll;
1947 //              res.find(H(wild(1),wild(2)), ll);
1948 //              res.find(zeta(wild(1)), ll);
1949 //              res.find(zeta(wild(1),wild(2)), ll);
1950 //              res = res.collect(ll);
1951
1952                 return res.subs(xtemp == numeric(x)).evalf();
1953         }
1954
1955         return H(x1,x2).hold();
1956 }
1957
1958
1959 static ex H_eval(const ex& m_, const ex& x)
1960 {
1961         lst m;
1962         if (is_a<lst>(m_)) {
1963                 m = ex_to<lst>(m_);
1964         } else {
1965                 m = lst(m_);
1966         }
1967         if (m.nops() == 0) {
1968                 return _ex1;
1969         }
1970         ex pos1;
1971         ex pos2;
1972         ex n;
1973         ex p;
1974         int step = 0;
1975         if (*m.begin() > _ex1) {
1976                 step++;
1977                 pos1 = _ex0;
1978                 pos2 = _ex1;
1979                 n = *m.begin()-1;
1980                 p = _ex1;
1981         } else if (*m.begin() < _ex_1) {
1982                 step++;
1983                 pos1 = _ex0;
1984                 pos2 = _ex_1;
1985                 n = -*m.begin()-1;
1986                 p = _ex1;
1987         } else if (*m.begin() == _ex0) {
1988                 pos1 = _ex0;
1989                 n = _ex1;
1990         } else {
1991                 pos1 = *m.begin();
1992                 p = _ex1;
1993         }
1994         for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
1995                 if ((*it).info(info_flags::integer)) {
1996                         if (step == 0) {
1997                                 if (*it > _ex1) {
1998                                         if (pos1 == _ex0) {
1999                                                 step = 1;
2000                                                 pos2 = _ex1;
2001                                                 n += *it-1;
2002                                                 p = _ex1;
2003                                         } else {
2004                                                 step = 2;
2005                                         }
2006                                 } else if (*it < _ex_1) {
2007                                         if (pos1 == _ex0) {
2008                                                 step = 1;
2009                                                 pos2 = _ex_1;
2010                                                 n += -*it-1;
2011                                                 p = _ex1;
2012                                         } else {
2013                                                 step = 2;
2014                                         }
2015                                 } else {
2016                                         if (*it != pos1) {
2017                                                 step = 1;
2018                                                 pos2 = *it;
2019                                         }
2020                                         if (*it == _ex0) {
2021                                                 n++;
2022                                         } else {
2023                                                 p++;
2024                                         }
2025                                 }
2026                         } else if (step == 1) {
2027                                 if (*it != pos2) {
2028                                         step = 2;
2029                                 } else {
2030                                         if (*it == _ex0) {
2031                                                 n++;
2032                                         } else {
2033                                                 p++;
2034                                         }
2035                                 }
2036                         }
2037                 } else {
2038                         // if some m_i is not an integer
2039                         return H(m_, x).hold();
2040                 }
2041         }
2042         if ((x == _ex1) && (*(--m.end()) != _ex0)) {
2043                 return convert_H_to_zeta(m);
2044         }
2045         if (step == 0) {
2046                 if (pos1 == _ex0) {
2047                         // all zero
2048                         if (x == _ex0) {
2049                                 return H(m_, x).hold();
2050                         }
2051                         return pow(log(x), m.nops()) / factorial(m.nops());
2052                 } else {
2053                         // all (minus) one
2054                         return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
2055                 }
2056         } else if ((step == 1) && (pos1 == _ex0)){
2057                 // convertible to S
2058                 if (pos2 == _ex1) {
2059                         return S(n, p, x);
2060                 } else {
2061                         return pow(-1, p) * S(n, p, -x);
2062                 }
2063         }
2064         if (x == _ex0) {
2065                 return _ex0;
2066         }
2067         if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2068                 return H(m_, x).evalf();
2069         }
2070         return H(m_, x).hold();
2071 }
2072
2073
2074 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
2075 {
2076         epvector seq;
2077         seq.push_back(expair(H(m, x), 0));
2078         return pseries(rel, seq);
2079 }
2080
2081
2082 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
2083 {
2084         GINAC_ASSERT(deriv_param < 2);
2085         if (deriv_param == 0) {
2086                 return _ex0;
2087         }
2088         lst m;
2089         if (is_a<lst>(m_)) {
2090                 m = ex_to<lst>(m_);
2091         } else {
2092                 m = lst(m_);
2093         }
2094         ex mb = *m.begin();
2095         if (mb > _ex1) {
2096                 m[0]--;
2097                 return H(m, x) / x;
2098         }
2099         if (mb < _ex_1) {
2100                 m[0]++;
2101                 return H(m, x) / x;
2102         }
2103         m.remove_first();
2104         if (mb == _ex1) {
2105                 return 1/(1-x) * H(m, x);
2106         } else if (mb == _ex_1) {
2107                 return 1/(1+x) * H(m, x);
2108         } else {
2109                 return H(m, x) / x;
2110         }
2111 }
2112
2113
2114 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
2115 {
2116         lst m;
2117         if (is_a<lst>(m_)) {
2118                 m = ex_to<lst>(m_);
2119         } else {
2120                 m = lst(m_);
2121         }
2122         c.s << "\\mbox{H}_{";
2123         lst::const_iterator itm = m.begin();
2124         (*itm).print(c);
2125         itm++;
2126         for (; itm != m.end(); itm++) {
2127                 c.s << ",";
2128                 (*itm).print(c);
2129         }
2130         c.s << "}(";
2131         x.print(c);
2132         c.s << ")";
2133 }
2134
2135
2136 REGISTER_FUNCTION(H,
2137                   evalf_func(H_evalf).
2138                   eval_func(H_eval).
2139                   series_func(H_series).
2140                   derivative_func(H_deriv).
2141                   print_func<print_latex>(H_print_latex).
2142                   do_not_evalf_params());
2143
2144
2145 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
2146 ex convert_H_to_Li(const ex& m, const ex& x)
2147 {
2148         map_trafo_H_reduce_trailing_zeros filter;
2149         map_trafo_H_convert_to_Li filter2;
2150         if (is_a<lst>(m)) {
2151                 return filter2(filter(H(m, x).hold()));
2152         } else {
2153                 return filter2(filter(H(lst(m), x).hold()));
2154         }
2155 }
2156
2157
2158 //////////////////////////////////////////////////////////////////////
2159 //
2160 // Multiple zeta values  zeta(x) and zeta(x,s)
2161 //
2162 // helper functions
2163 //
2164 //////////////////////////////////////////////////////////////////////
2165
2166
2167 // anonymous namespace for helper functions
2168 namespace {
2169
2170
2171 // parameters and data for [Cra] algorithm
2172 const cln::cl_N lambda = cln::cl_N("319/320");
2173 int L1;
2174 int L2;
2175 std::vector<std::vector<cln::cl_N> > f_kj;
2176 std::vector<cln::cl_N> crB;
2177 std::vector<std::vector<cln::cl_N> > crG;
2178 std::vector<cln::cl_N> crX;
2179
2180
2181 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
2182 {
2183         const int size = a.size();
2184         for (int n=0; n<size; n++) {
2185                 c[n] = 0;
2186                 for (int m=0; m<=n; m++) {
2187                         c[n] = c[n] + a[m]*b[n-m];
2188                 }
2189         }
2190 }
2191
2192
2193 // [Cra] section 4
2194 void initcX(const std::vector<int>& s)
2195 {
2196         const int k = s.size();
2197
2198         crX.clear();
2199         crG.clear();
2200         crB.clear();
2201
2202         for (int i=0; i<=L2; i++) {
2203                 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
2204         }
2205
2206         int Sm = 0;
2207         int Smp1 = 0;
2208         for (int m=0; m<k-1; m++) {
2209                 std::vector<cln::cl_N> crGbuf;
2210                 Sm = Sm + s[m];
2211                 Smp1 = Sm + s[m+1];
2212                 for (int i=0; i<=L2; i++) {
2213                         crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
2214                 }
2215                 crG.push_back(crGbuf);
2216         }
2217
2218         crX = crB;
2219
2220         for (int m=0; m<k-1; m++) {
2221                 std::vector<cln::cl_N> Xbuf;
2222                 for (int i=0; i<=L2; i++) {
2223                         Xbuf.push_back(crX[i] * crG[m][i]);
2224                 }
2225                 halfcyclic_convolute(Xbuf, crB, crX);
2226         }
2227 }
2228
2229
2230 // [Cra] section 4
2231 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
2232 {
2233         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2234         cln::cl_N factor = cln::expt(lambda, Sqk);
2235         cln::cl_N res = factor / Sqk * crX[0] * one;
2236         cln::cl_N resbuf;
2237         int N = 0;
2238         do {
2239                 resbuf = res;
2240                 factor = factor * lambda;
2241                 N++;
2242                 res = res + crX[N] * factor / (N+Sqk);
2243         } while ((res != resbuf) || cln::zerop(crX[N]));
2244         return res;
2245 }
2246
2247
2248 // [Cra] section 4
2249 void calc_f(int maxr)
2250 {
2251         f_kj.clear();
2252         f_kj.resize(L1);
2253         
2254         cln::cl_N t0, t1, t2, t3, t4;
2255         int i, j, k;
2256         std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
2257         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2258         
2259         t0 = cln::exp(-lambda);
2260         t2 = 1;
2261         for (k=1; k<=L1; k++) {
2262                 t1 = k * lambda;
2263                 t2 = t0 * t2;
2264                 for (j=1; j<=maxr; j++) {
2265                         t3 = 1;
2266                         t4 = 1;
2267                         for (i=2; i<=j; i++) {
2268                                 t4 = t4 * (j-i+1);
2269                                 t3 = t1 * t3 + t4;
2270                         }
2271                         (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
2272                 }
2273                 it++;
2274         }
2275 }
2276
2277
2278 // [Cra] (3.1)
2279 cln::cl_N crandall_Z(const std::vector<int>& s)
2280 {
2281         const int j = s.size();
2282
2283         if (j == 1) {   
2284                 cln::cl_N t0;
2285                 cln::cl_N t0buf;
2286                 int q = 0;
2287                 do {
2288                         t0buf = t0;
2289                         q++;
2290                         t0 = t0 + f_kj[q+j-2][s[0]-1];
2291                 } while (t0 != t0buf);
2292                 
2293                 return t0 / cln::factorial(s[0]-1);
2294         }
2295
2296         std::vector<cln::cl_N> t(j);
2297
2298         cln::cl_N t0buf;
2299         int q = 0;
2300         do {
2301                 t0buf = t[0];
2302                 q++;
2303                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
2304                 for (int k=j-2; k>=1; k--) {
2305                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
2306                 }
2307                 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
2308         } while (t[0] != t0buf);
2309         
2310         return t[0] / cln::factorial(s[0]-1);
2311 }
2312
2313
2314 // [Cra] (2.4)
2315 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
2316 {
2317         std::vector<int> r = s;
2318         const int j = r.size();
2319
2320         // decide on maximal size of f_kj for crandall_Z
2321         if (Digits < 50) {
2322                 L1 = 150;
2323         } else {
2324                 L1 = Digits * 3 + j*2;
2325         }
2326
2327         // decide on maximal size of crX for crandall_Y
2328         if (Digits < 38) {
2329                 L2 = 63;
2330         } else if (Digits < 86) {
2331                 L2 = 127;
2332         } else if (Digits < 192) {
2333                 L2 = 255;
2334         } else if (Digits < 394) {
2335                 L2 = 511;
2336         } else if (Digits < 808) {
2337                 L2 = 1023;
2338         } else {
2339                 L2 = 2047;
2340         }
2341
2342         cln::cl_N res;
2343
2344         int maxr = 0;
2345         int S = 0;
2346         for (int i=0; i<j; i++) {
2347                 S += r[i];
2348                 if (r[i] > maxr) {
2349                         maxr = r[i];
2350                 }
2351         }
2352
2353         calc_f(maxr);
2354
2355         const cln::cl_N r0factorial = cln::factorial(r[0]-1);
2356
2357         std::vector<int> rz;
2358         int skp1buf;
2359         int Srun = S;
2360         for (int k=r.size()-1; k>0; k--) {
2361
2362                 rz.insert(rz.begin(), r.back());
2363                 skp1buf = rz.front();
2364                 Srun -= skp1buf;
2365                 r.pop_back();
2366
2367                 initcX(r);
2368                 
2369                 for (int q=0; q<skp1buf; q++) {
2370                         
2371                         cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2372                         cln::cl_N pp2 = crandall_Z(rz);
2373
2374                         rz.front()--;
2375                         
2376                         if (q & 1) {
2377                                 res = res - pp1 * pp2 / cln::factorial(q);
2378                         } else {
2379                                 res = res + pp1 * pp2 / cln::factorial(q);
2380                         }
2381                 }
2382                 rz.front() = skp1buf;
2383         }
2384         rz.insert(rz.begin(), r.back());
2385
2386         initcX(rz);
2387
2388         res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2389
2390         return res;
2391 }
2392
2393
2394 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2395 {
2396         const int j = r.size();
2397
2398         // buffer for subsums
2399         std::vector<cln::cl_N> t(j);
2400         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2401
2402         cln::cl_N t0buf;
2403         int q = 0;
2404         do {
2405                 t0buf = t[0];
2406                 q++;
2407                 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2408                 for (int k=j-2; k>=0; k--) {
2409                         t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2410                 }
2411         } while (t[0] != t0buf);
2412
2413         return t[0];
2414 }
2415
2416
2417 // does Hoelder convolution. see [BBB] (7.0)
2418 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2419 {
2420         // prepare parameters
2421         // holds Li arguments in [BBB] notation
2422         std::vector<int> s = s_;
2423         std::vector<int> m_p = m_;
2424         std::vector<int> m_q;
2425         // holds Li arguments in nested sums notation
2426         std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2427         s_p[0] = s_p[0] * cln::cl_N("1/2");
2428         // convert notations
2429         int sig = 1;
2430         for (int i=0; i<s_.size(); i++) {
2431                 if (s_[i] < 0) {
2432                         sig = -sig;
2433                         s_p[i] = -s_p[i];
2434                 }
2435                 s[i] = sig * std::abs(s[i]);
2436         }
2437         std::vector<cln::cl_N> s_q;
2438         cln::cl_N signum = 1;
2439
2440         // first term
2441         cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2442
2443         // middle terms
2444         do {
2445
2446                 // change parameters
2447                 if (s.front() > 0) {
2448                         if (m_p.front() == 1) {
2449                                 m_p.erase(m_p.begin());
2450                                 s_p.erase(s_p.begin());
2451                                 if (s_p.size() > 0) {
2452                                         s_p.front() = s_p.front() * cln::cl_N("1/2");
2453                                 }
2454                                 s.erase(s.begin());
2455                                 m_q.front()++;
2456                         } else {
2457                                 m_p.front()--;
2458                                 m_q.insert(m_q.begin(), 1);
2459                                 if (s_q.size() > 0) {
2460                                         s_q.front() = s_q.front() * 2;
2461                                 }
2462                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2463                         }
2464                 } else {
2465                         if (m_p.front() == 1) {
2466                                 m_p.erase(m_p.begin());
2467                                 cln::cl_N spbuf = s_p.front();
2468                                 s_p.erase(s_p.begin());
2469                                 if (s_p.size() > 0) {
2470                                         s_p.front() = s_p.front() * spbuf;
2471                                 }
2472                                 s.erase(s.begin());
2473                                 m_q.insert(m_q.begin(), 1);
2474                                 if (s_q.size() > 0) {
2475                                         s_q.front() = s_q.front() * 4;
2476                                 }
2477                                 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2478                                 signum = -signum;
2479                         } else {
2480                                 m_p.front()--;
2481                                 m_q.insert(m_q.begin(), 1);
2482                                 if (s_q.size() > 0) {
2483                                         s_q.front() = s_q.front() * 2;
2484                                 }
2485                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2486                         }
2487                 }
2488
2489                 // exiting the loop
2490                 if (m_p.size() == 0) break;
2491
2492                 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2493
2494         } while (true);
2495
2496         // last term
2497         res = res + signum * multipleLi_do_sum(m_q, s_q);
2498
2499         return res;
2500 }
2501
2502
2503 } // end of anonymous namespace
2504
2505
2506 //////////////////////////////////////////////////////////////////////
2507 //
2508 // Multiple zeta values  zeta(x)
2509 //
2510 // GiNaC function
2511 //
2512 //////////////////////////////////////////////////////////////////////
2513
2514
2515 static ex zeta1_evalf(const ex& x)
2516 {
2517         if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2518
2519                 // multiple zeta value
2520                 const int count = x.nops();
2521                 const lst& xlst = ex_to<lst>(x);
2522                 std::vector<int> r(count);
2523
2524                 // check parameters and convert them
2525                 lst::const_iterator it1 = xlst.begin();
2526                 std::vector<int>::iterator it2 = r.begin();
2527                 do {
2528                         if (!(*it1).info(info_flags::posint)) {
2529                                 return zeta(x).hold();
2530                         }
2531                         *it2 = ex_to<numeric>(*it1).to_int();
2532                         it1++;
2533                         it2++;
2534                 } while (it2 != r.end());
2535
2536                 // check for divergence
2537                 if (r[0] == 1) {
2538                         return zeta(x).hold();
2539                 }
2540
2541                 // decide on summation algorithm
2542                 // this is still a bit clumsy
2543                 int limit = (Digits>17) ? 10 : 6;
2544                 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2545                         return numeric(zeta_do_sum_Crandall(r));
2546                 } else {
2547                         return numeric(zeta_do_sum_simple(r));
2548                 }
2549         }
2550
2551         // single zeta value
2552         if (is_exactly_a<numeric>(x) && (x != 1)) {
2553                 try {
2554                         return zeta(ex_to<numeric>(x));
2555                 } catch (const dunno &e) { }
2556         }
2557
2558         return zeta(x).hold();
2559 }
2560
2561
2562 static ex zeta1_eval(const ex& m)
2563 {
2564         if (is_exactly_a<lst>(m)) {
2565                 if (m.nops() == 1) {
2566                         return zeta(m.op(0));
2567                 }
2568                 return zeta(m).hold();
2569         }
2570
2571         if (m.info(info_flags::numeric)) {
2572                 const numeric& y = ex_to<numeric>(m);
2573                 // trap integer arguments:
2574                 if (y.is_integer()) {
2575                         if (y.is_zero()) {
2576                                 return _ex_1_2;
2577                         }
2578                         if (y.is_equal(_num1)) {
2579                                 return zeta(m).hold();
2580                         }
2581                         if (y.info(info_flags::posint)) {
2582                                 if (y.info(info_flags::odd)) {
2583                                         return zeta(m).hold();
2584                                 } else {
2585                                         return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2586                                 }
2587                         } else {
2588                                 if (y.info(info_flags::odd)) {
2589                                         return -bernoulli(_num1-y) / (_num1-y);
2590                                 } else {
2591                                         return _ex0;
2592                                 }
2593                         }
2594                 }
2595                 // zeta(float)
2596                 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
2597                         return zeta1_evalf(m);
2598                 }
2599         }
2600         return zeta(m).hold();
2601 }
2602
2603
2604 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
2605 {
2606         GINAC_ASSERT(deriv_param==0);
2607
2608         if (is_exactly_a<lst>(m)) {
2609                 return _ex0;
2610         } else {
2611                 return zetaderiv(_ex1, m);
2612         }
2613 }
2614
2615
2616 static void zeta1_print_latex(const ex& m_, const print_context& c)
2617 {
2618         c.s << "\\zeta(";
2619         if (is_a<lst>(m_)) {
2620                 const lst& m = ex_to<lst>(m_);
2621                 lst::const_iterator it = m.begin();
2622                 (*it).print(c);
2623                 it++;
2624                 for (; it != m.end(); it++) {
2625                         c.s << ",";
2626                         (*it).print(c);
2627                 }
2628         } else {
2629                 m_.print(c);
2630         }
2631         c.s << ")";
2632 }
2633
2634
2635 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta").
2636                                 evalf_func(zeta1_evalf).
2637                                 eval_func(zeta1_eval).
2638                                 derivative_func(zeta1_deriv).
2639                                 print_func<print_latex>(zeta1_print_latex).
2640                                 do_not_evalf_params().
2641                                 overloaded(2));
2642
2643
2644 //////////////////////////////////////////////////////////////////////
2645 //
2646 // Alternating Euler sum  zeta(x,s)
2647 //
2648 // GiNaC function
2649 //
2650 //////////////////////////////////////////////////////////////////////
2651
2652
2653 static ex zeta2_evalf(const ex& x, const ex& s)
2654 {
2655         if (is_exactly_a<lst>(x)) {
2656
2657                 // alternating Euler sum
2658                 const int count = x.nops();
2659                 const lst& xlst = ex_to<lst>(x);
2660                 const lst& slst = ex_to<lst>(s);
2661                 std::vector<int> xi(count);
2662                 std::vector<int> si(count);
2663
2664                 // check parameters and convert them
2665                 lst::const_iterator it_xread = xlst.begin();
2666                 lst::const_iterator it_sread = slst.begin();
2667                 std::vector<int>::iterator it_xwrite = xi.begin();
2668                 std::vector<int>::iterator it_swrite = si.begin();
2669                 do {
2670                         if (!(*it_xread).info(info_flags::posint)) {
2671                                 return zeta(x, s).hold();
2672                         }
2673                         *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2674                         if (*it_sread > 0) {
2675                                 *it_swrite = 1;
2676                         } else {
2677                                 *it_swrite = -1;
2678                         }
2679                         it_xread++;
2680                         it_sread++;
2681                         it_xwrite++;
2682                         it_swrite++;
2683                 } while (it_xwrite != xi.end());
2684
2685                 // check for divergence
2686                 if ((xi[0] == 1) && (si[0] == 1)) {
2687                         return zeta(x, s).hold();
2688                 }
2689
2690                 // use Hoelder convolution
2691                 return numeric(zeta_do_Hoelder_convolution(xi, si));
2692         }
2693
2694         return zeta(x, s).hold();
2695 }
2696
2697
2698 static ex zeta2_eval(const ex& m, const ex& s_)
2699 {
2700         if (is_exactly_a<lst>(s_)) {
2701                 const lst& s = ex_to<lst>(s_);
2702                 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
2703                         if ((*it).info(info_flags::positive)) {
2704                                 continue;
2705                         }
2706                         return zeta(m, s_).hold();
2707                 }
2708                 return zeta(m);
2709         } else if (s_.info(info_flags::positive)) {
2710                 return zeta(m);
2711         }
2712
2713         return zeta(m, s_).hold();
2714 }
2715
2716
2717 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
2718 {
2719         GINAC_ASSERT(deriv_param==0);
2720
2721         if (is_exactly_a<lst>(m)) {
2722                 return _ex0;
2723         } else {
2724                 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
2725                         return zetaderiv(_ex1, m);
2726                 }
2727                 return _ex0;
2728         }
2729 }
2730
2731
2732 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
2733 {
2734         lst m;
2735         if (is_a<lst>(m_)) {
2736                 m = ex_to<lst>(m_);
2737         } else {
2738                 m = lst(m_);
2739         }
2740         lst s;
2741         if (is_a<lst>(s_)) {
2742                 s = ex_to<lst>(s_);
2743         } else {
2744                 s = lst(s_);
2745         }
2746         c.s << "\\zeta(";
2747         lst::const_iterator itm = m.begin();
2748         lst::const_iterator its = s.begin();
2749         if (*its < 0) {
2750                 c.s << "\\overline{";
2751                 (*itm).print(c);
2752                 c.s << "}";
2753         } else {
2754                 (*itm).print(c);
2755         }
2756         its++;
2757         itm++;
2758         for (; itm != m.end(); itm++, its++) {
2759                 c.s << ",";
2760                 if (*its < 0) {
2761                         c.s << "\\overline{";
2762                         (*itm).print(c);
2763                         c.s << "}";
2764                 } else {
2765                         (*itm).print(c);
2766                 }
2767         }
2768         c.s << ")";
2769 }
2770
2771
2772 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta").
2773                                 evalf_func(zeta2_evalf).
2774                                 eval_func(zeta2_eval).
2775                                 derivative_func(zeta2_deriv).
2776                                 print_func<print_latex>(zeta2_print_latex).
2777                                 do_not_evalf_params().
2778                                 overloaded(2));
2779
2780
2781 } // namespace GiNaC
2782