S(0,p,x) now evaluates numerically
[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         // [Kol] (5.3)
943         if ((cln::realpart(value) < -0.5) || (n == 0)) {
944
945                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
946                                    * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
947
948                 for (int s=0; s<n; s++) {
949                         cln::cl_N res2;
950                         for (int r=0; r<p; r++) {
951                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
952                                               * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
953                         }
954                         result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
955                 }
956
957                 return result;
958                 
959         }
960         // [Kol] (5.12)
961         if (cln::abs(value) > 1) {
962                 
963                 cln::cl_N result;
964
965                 for (int s=0; s<p; s++) {
966                         for (int r=0; r<=s; r++) {
967                                 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
968                                                   / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
969                                                   * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
970                         }
971                 }
972                 result = result * cln::expt(cln::cl_I(-1),n);
973
974                 cln::cl_N res2;
975                 for (int r=0; r<n; r++) {
976                         res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
977                 }
978                 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
979
980                 result = result + cln::expt(cln::cl_I(-1),p) * res2;
981
982                 return result;
983         }
984         else {
985                 return S_projection(n, p, value, prec);
986         }
987 }
988
989
990 } // end of anonymous namespace
991
992
993 //////////////////////////////////////////////////////////////////////
994 //
995 // Nielsen's generalized polylogarithm  S(n,p,x)
996 //
997 // GiNaC function
998 //
999 //////////////////////////////////////////////////////////////////////
1000
1001
1002 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1003 {
1004         if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a<numeric>(x)) {
1005                 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1006         }
1007         return S(n, p, x).hold();
1008 }
1009
1010
1011 static ex S_eval(const ex& n, const ex& p, const ex& x)
1012 {
1013         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1014                 if (x == 0) {
1015                         return _ex0;
1016                 }
1017                 if (x == 1) {
1018                         lst m(n+1);
1019                         for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
1020                                 m.append(1);
1021                         }
1022                         return zeta(m);
1023                 }
1024                 if (p == 1) {
1025                         return Li(n+1, x);
1026                 }
1027                 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
1028                         return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1029                 }
1030         }
1031         if (n.is_zero()) {
1032                 // [Kol] (5.3)
1033                 return pow(-log(1-x), p) / factorial(p);
1034         }
1035         return S(n, p, x).hold();
1036 }
1037
1038
1039 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
1040 {
1041         epvector seq;
1042         seq.push_back(expair(S(n, p, x), 0));
1043         return pseries(rel, seq);
1044 }
1045
1046
1047 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
1048 {
1049         GINAC_ASSERT(deriv_param < 3);
1050         if (deriv_param < 2) {
1051                 return _ex0;
1052         }
1053         if (n > 0) {
1054                 return S(n-1, p, x) / x;
1055         } else {
1056                 return S(n, p-1, x) / (1-x);
1057         }
1058 }
1059
1060
1061 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
1062 {
1063         c.s << "\\mbox{S}_{";
1064         n.print(c);
1065         c.s << ",";
1066         p.print(c);
1067         c.s << "}(";
1068         x.print(c);
1069         c.s << ")";
1070 }
1071
1072
1073 REGISTER_FUNCTION(S,
1074                   evalf_func(S_evalf).
1075                   eval_func(S_eval).
1076                   series_func(S_series).
1077                   derivative_func(S_deriv).
1078                   print_func<print_latex>(S_print_latex).
1079                   do_not_evalf_params());
1080
1081
1082 //////////////////////////////////////////////////////////////////////
1083 //
1084 // Harmonic polylogarithm  H(m,x)
1085 //
1086 // helper functions
1087 //
1088 //////////////////////////////////////////////////////////////////////
1089
1090
1091 // anonymous namespace for helper functions
1092 namespace {
1093
1094         
1095 // regulates the pole (used by 1/x-transformation)
1096 symbol H_polesign("IMSIGN");
1097
1098
1099 // convert parameters from H to Li representation
1100 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
1101 // returns true if some parameters are negative
1102 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
1103 {
1104         // expand parameter list
1105         lst mexp;
1106         for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
1107                 if (*it > 1) {
1108                         for (ex count=*it-1; count > 0; count--) {
1109                                 mexp.append(0);
1110                         }
1111                         mexp.append(1);
1112                 } else if (*it < -1) {
1113                         for (ex count=*it+1; count < 0; count++) {
1114                                 mexp.append(0);
1115                         }
1116                         mexp.append(-1);
1117                 } else {
1118                         mexp.append(*it);
1119                 }
1120         }
1121         
1122         ex signum = 1;
1123         pf = 1;
1124         bool has_negative_parameters = false;
1125         ex acc = 1;
1126         for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
1127                 if (*it == 0) {
1128                         acc++;
1129                         continue;
1130                 }
1131                 if (*it > 0) {
1132                         m.append((*it+acc-1) * signum);
1133                 } else {
1134                         m.append((*it-acc+1) * signum);
1135                 }
1136                 acc = 1;
1137                 signum = *it;
1138                 pf *= *it;
1139                 if (pf < 0) {
1140                         has_negative_parameters = true;
1141                 }
1142         }
1143         if (has_negative_parameters) {
1144                 for (int i=0; i<m.nops(); i++) {
1145                         if (m.op(i) < 0) {
1146                                 m.let_op(i) = -m.op(i);
1147                                 s.append(-1);
1148                         } else {
1149                                 s.append(1);
1150                         }
1151                 }
1152         }
1153         
1154         return has_negative_parameters;
1155 }
1156
1157
1158 // recursivly transforms H to corresponding multiple polylogarithms
1159 struct map_trafo_H_convert_to_Li : public map_function
1160 {
1161         ex operator()(const ex& e)
1162         {
1163                 if (is_a<add>(e) || is_a<mul>(e)) {
1164                         return e.map(*this);
1165                 }
1166                 if (is_a<function>(e)) {
1167                         std::string name = ex_to<function>(e).get_name();
1168                         if (name == "H") {
1169                                 lst parameter;
1170                                 if (is_a<lst>(e.op(0))) {
1171                                                 parameter = ex_to<lst>(e.op(0));
1172                                 } else {
1173                                         parameter = lst(e.op(0));
1174                                 }
1175                                 ex arg = e.op(1);
1176
1177                                 lst m;
1178                                 lst s;
1179                                 ex pf;
1180                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1181                                         s.let_op(0) = s.op(0) * arg;
1182                                         return pf * Li(m, s).hold();
1183                                 } else {
1184                                         for (int i=0; i<m.nops(); i++) {
1185                                                 s.append(1);
1186                                         }
1187                                         s.let_op(0) = s.op(0) * arg;
1188                                         return Li(m, s).hold();
1189                                 }
1190                         }
1191                 }
1192                 return e;
1193         }
1194 };
1195
1196
1197 // recursivly transforms H to corresponding zetas
1198 struct map_trafo_H_convert_to_zeta : public map_function
1199 {
1200         ex operator()(const ex& e)
1201         {
1202                 if (is_a<add>(e) || is_a<mul>(e)) {
1203                         return e.map(*this);
1204                 }
1205                 if (is_a<function>(e)) {
1206                         std::string name = ex_to<function>(e).get_name();
1207                         if (name == "H") {
1208                                 lst parameter;
1209                                 if (is_a<lst>(e.op(0))) {
1210                                                 parameter = ex_to<lst>(e.op(0));
1211                                 } else {
1212                                         parameter = lst(e.op(0));
1213                                 }
1214
1215                                 lst m;
1216                                 lst s;
1217                                 ex pf;
1218                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1219                                         return pf * zeta(m, s);
1220                                 } else {
1221                                         return zeta(m);
1222                                 }
1223                         }
1224                 }
1225                 return e;
1226         }
1227 };
1228
1229
1230 // remove trailing zeros from H-parameters
1231 struct map_trafo_H_reduce_trailing_zeros : public map_function
1232 {
1233         ex operator()(const ex& e)
1234         {
1235                 if (is_a<add>(e) || is_a<mul>(e)) {
1236                         return e.map(*this);
1237                 }
1238                 if (is_a<function>(e)) {
1239                         std::string name = ex_to<function>(e).get_name();
1240                         if (name == "H") {
1241                                 lst parameter;
1242                                 if (is_a<lst>(e.op(0))) {
1243                                         parameter = ex_to<lst>(e.op(0));
1244                                 } else {
1245                                         parameter = lst(e.op(0));
1246                                 }
1247                                 ex arg = e.op(1);
1248                                 if (parameter.op(parameter.nops()-1) == 0) {
1249                                         
1250                                         //
1251                                         if (parameter.nops() == 1) {
1252                                                 return log(arg);
1253                                         }
1254                                         
1255                                         //
1256                                         lst::const_iterator it = parameter.begin();
1257                                         while ((it != parameter.end()) && (*it == 0)) {
1258                                                 it++;
1259                                         }
1260                                         if (it == parameter.end()) {
1261                                                 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1262                                         }
1263                                         
1264                                         //
1265                                         parameter.remove_last();
1266                                         int lastentry = parameter.nops();
1267                                         while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1268                                                 lastentry--;
1269                                         }
1270                                         
1271                                         //
1272                                         ex result = log(arg) * H(parameter,arg).hold();
1273                                         ex acc = 0;
1274                                         for (ex i=0; i<lastentry; i++) {
1275                                                 if (parameter[i] > 0) {
1276                                                         parameter[i]++;
1277                                                         result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
1278                                                         parameter[i]--;
1279                                                         acc = 0;
1280                                                 } else if (parameter[i] < 0) {
1281                                                         parameter[i]--;
1282                                                         result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
1283                                                         parameter[i]++;
1284                                                         acc = 0;
1285                                                 } else {
1286                                                         acc++;
1287                                                 }
1288                                         }
1289                                         
1290                                         if (lastentry < parameter.nops()) {
1291                                                 result = result / (parameter.nops()-lastentry+1);
1292                                                 return result.map(*this);
1293                                         } else {
1294                                                 return result;
1295                                         }
1296                                 }
1297                         }
1298                 }
1299                 return e;
1300         }
1301 };
1302
1303
1304 // returns an expression with zeta functions corresponding to the parameter list for H
1305 ex convert_H_to_zeta(const lst& m)
1306 {
1307         symbol xtemp("xtemp");
1308         map_trafo_H_reduce_trailing_zeros filter;
1309         map_trafo_H_convert_to_zeta filter2;
1310         return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
1311 }
1312
1313
1314 // convert signs form Li to H representation
1315 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
1316 {
1317         lst res;
1318         lst::const_iterator itm = m.begin();
1319         lst::const_iterator itx = ++x.begin();
1320         ex signum = _ex1;
1321         pf = _ex1;
1322         res.append(*itm);
1323         itm++;
1324         while (itx != x.end()) {
1325                 signum *= *itx;
1326                 pf *= signum;
1327                 res.append((*itm) * signum);
1328                 itm++;
1329                 itx++;
1330         }
1331         return res;
1332 }
1333
1334
1335 // multiplies an one-dimensional H with another H
1336 // [ReV] (18)
1337 ex trafo_H_mult(const ex& h1, const ex& h2)
1338 {
1339         ex res;
1340         ex hshort;
1341         lst hlong;
1342         ex h1nops = h1.op(0).nops();
1343         ex h2nops = h2.op(0).nops();
1344         if (h1nops > 1) {
1345                 hshort = h2.op(0).op(0);
1346                 hlong = ex_to<lst>(h1.op(0));
1347         } else {
1348                 hshort = h1.op(0).op(0);
1349                 if (h2nops > 1) {
1350                         hlong = ex_to<lst>(h2.op(0));
1351                 } else {
1352                         hlong = h2.op(0).op(0);
1353                 }
1354         }
1355         for (int i=0; i<=hlong.nops(); i++) {
1356                 lst newparameter;
1357                 int j=0;
1358                 for (; j<i; j++) {
1359                         newparameter.append(hlong[j]);
1360                 }
1361                 newparameter.append(hshort);
1362                 for (; j<hlong.nops(); j++) {
1363                         newparameter.append(hlong[j]);
1364                 }
1365                 res += H(newparameter, h1.op(1)).hold();
1366         }
1367         return res;
1368 }
1369
1370
1371 // applies trafo_H_mult recursively on expressions
1372 struct map_trafo_H_mult : public map_function
1373 {
1374         ex operator()(const ex& e)
1375         {
1376                 if (is_a<add>(e)) {
1377                         return e.map(*this);
1378                 }
1379
1380                 if (is_a<mul>(e)) {
1381
1382                         ex result = 1;
1383                         ex firstH;
1384                         lst Hlst;
1385                         for (int pos=0; pos<e.nops(); pos++) {
1386                                 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1387                                         std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1388                                         if (name == "H") {
1389                                                 for (ex i=0; i<e.op(pos).op(1); i++) {
1390                                                         Hlst.append(e.op(pos).op(0));
1391                                                 }
1392                                                 continue;
1393                                         }
1394                                 } else if (is_a<function>(e.op(pos))) {
1395                                         std::string name = ex_to<function>(e.op(pos)).get_name();
1396                                         if (name == "H") {
1397                                                 if (e.op(pos).op(0).nops() > 1) {
1398                                                         firstH = e.op(pos);
1399                                                 } else {
1400                                                         Hlst.append(e.op(pos));
1401                                                 }
1402                                                 continue;
1403                                         }
1404                                 }
1405                                 result *= e.op(pos);
1406                         }
1407                         if (firstH == 0) {
1408                                 if (Hlst.nops() > 0) {
1409                                         firstH = Hlst[Hlst.nops()-1];
1410                                         Hlst.remove_last();
1411                                 } else {
1412                                         return e;
1413                                 }
1414                         }
1415
1416                         if (Hlst.nops() > 0) {
1417                                 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1418                                 result *= buffer;
1419                                 for (int i=1; i<Hlst.nops(); i++) {
1420                                         result *= Hlst.op(i);
1421                                 }
1422                                 result = result.expand();
1423                                 map_trafo_H_mult recursion;
1424                                 return recursion(result);
1425                         } else {
1426                                 return e;
1427                         }
1428
1429                 }
1430                 return e;
1431         }
1432 };
1433
1434
1435 // do integration [ReV] (55)
1436 // put parameter 0 in front of existing parameters
1437 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
1438 {
1439         ex h;
1440         std::string name;
1441         if (is_a<function>(e)) {
1442                 name = ex_to<function>(e).get_name();
1443         }
1444         if (name == "H") {
1445                 h = e;
1446         } else {
1447                 for (int i=0; i<e.nops(); i++) {
1448                         if (is_a<function>(e.op(i))) {
1449                                 std::string name = ex_to<function>(e.op(i)).get_name();
1450                                 if (name == "H") {
1451                                         h = e.op(i);
1452                                 }
1453                         }
1454                 }
1455         }
1456         if (h != 0) {
1457                 lst newparameter = ex_to<lst>(h.op(0));
1458                 newparameter.prepend(0);
1459                 ex addzeta = convert_H_to_zeta(newparameter);
1460                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1461         } else {
1462                 return e * (-H(lst(0),1/arg).hold());
1463         }
1464 }
1465
1466
1467 // do integration [ReV] (55)
1468 // put parameter -1 in front of existing parameters
1469 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
1470 {
1471         ex h;
1472         std::string name;
1473         if (is_a<function>(e)) {
1474                 name = ex_to<function>(e).get_name();
1475         }
1476         if (name == "H") {
1477                 h = e;
1478         } else {
1479                 for (int i=0; i<e.nops(); i++) {
1480                         if (is_a<function>(e.op(i))) {
1481                                 std::string name = ex_to<function>(e.op(i)).get_name();
1482                                 if (name == "H") {
1483                                         h = e.op(i);
1484                                 }
1485                         }
1486                 }
1487         }
1488         if (h != 0) {
1489                 lst newparameter = ex_to<lst>(h.op(0));
1490                 newparameter.prepend(-1);
1491                 ex addzeta = convert_H_to_zeta(newparameter);
1492                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1493         } else {
1494                 ex addzeta = convert_H_to_zeta(lst(-1));
1495                 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
1496         }
1497 }
1498
1499
1500 // do integration [ReV] (55)
1501 // put parameter -1 in front of existing parameters
1502 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
1503 {
1504         ex h;
1505         std::string name;
1506         if (is_a<function>(e)) {
1507                 name = ex_to<function>(e).get_name();
1508         }
1509         if (name == "H") {
1510                 h = e;
1511         } else {
1512                 for (int i=0; i<e.nops(); i++) {
1513                         if (is_a<function>(e.op(i))) {
1514                                 std::string name = ex_to<function>(e.op(i)).get_name();
1515                                 if (name == "H") {
1516                                         h = e.op(i);
1517                                 }
1518                         }
1519                 }
1520         }
1521         if (h != 0) {
1522                 lst newparameter = ex_to<lst>(h.op(0));
1523                 newparameter.prepend(-1);
1524                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1525         } else {
1526                 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
1527         }
1528 }
1529
1530
1531 // do integration [ReV] (55)
1532 // put parameter 1 in front of existing parameters
1533 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
1534 {
1535         ex h;
1536         std::string name;
1537         if (is_a<function>(e)) {
1538                 name = ex_to<function>(e).get_name();
1539         }
1540         if (name == "H") {
1541                 h = e;
1542         } else {
1543                 for (int i=0; i<e.nops(); i++) {
1544                         if (is_a<function>(e.op(i))) {
1545                                 std::string name = ex_to<function>(e.op(i)).get_name();
1546                                 if (name == "H") {
1547                                         h = e.op(i);
1548                                 }
1549                         }
1550                 }
1551         }
1552         if (h != 0) {
1553                 lst newparameter = ex_to<lst>(h.op(0));
1554                 newparameter.prepend(1);
1555                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1556         } else {
1557                 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
1558         }
1559 }
1560
1561
1562 // do x -> 1/x transformation
1563 struct map_trafo_H_1overx : public map_function
1564 {
1565         ex operator()(const ex& e)
1566         {
1567                 if (is_a<add>(e) || is_a<mul>(e)) {
1568                         return e.map(*this);
1569                 }
1570
1571                 if (is_a<function>(e)) {
1572                         std::string name = ex_to<function>(e).get_name();
1573                         if (name == "H") {
1574
1575                                 lst parameter = ex_to<lst>(e.op(0));
1576                                 ex arg = e.op(1);
1577
1578                                 // special cases if all parameters are either 0, 1 or -1
1579                                 bool allthesame = true;
1580                                 if (parameter.op(0) == 0) {
1581                                         for (int i=1; i<parameter.nops(); i++) {
1582                                                 if (parameter.op(i) != 0) {
1583                                                         allthesame = false;
1584                                                         break;
1585                                                 }
1586                                         }
1587                                         if (allthesame) {
1588                                                 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1589                                         }
1590                                 } else if (parameter.op(0) == -1) {
1591                                         for (int i=1; i<parameter.nops(); i++) {
1592                                                 if (parameter.op(i) != -1) {
1593                                                         allthesame = false;
1594                                                         break;
1595                                                 }
1596                                         }
1597                                         if (allthesame) {
1598                                                 map_trafo_H_mult unify;
1599                                                 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
1600                                                        / factorial(parameter.nops())).expand());
1601                                         }
1602                                 } else {
1603                                         for (int i=1; i<parameter.nops(); i++) {
1604                                                 if (parameter.op(i) != 1) {
1605                                                         allthesame = false;
1606                                                         break;
1607                                                 }
1608                                         }
1609                                         if (allthesame) {
1610                                                 map_trafo_H_mult unify;
1611                                                 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
1612                                                        / factorial(parameter.nops())).expand());
1613                                         }
1614                                 }
1615
1616                                 lst newparameter = parameter;
1617                                 newparameter.remove_first();
1618
1619                                 if (parameter.op(0) == 0) {
1620                                         
1621                                         // leading zero
1622                                         ex res = convert_H_to_zeta(parameter);
1623                                         map_trafo_H_1overx recursion;
1624                                         ex buffer = recursion(H(newparameter, arg).hold());
1625                                         if (is_a<add>(buffer)) {
1626                                                 for (int i=0; i<buffer.nops(); i++) {
1627                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
1628                                                 }
1629                                         } else {
1630                                                 res += trafo_H_1tx_prepend_zero(buffer, arg);
1631                                         }
1632                                         return res;
1633
1634                                 } else if (parameter.op(0) == -1) {
1635
1636                                         // leading negative one
1637                                         ex res = convert_H_to_zeta(parameter);
1638                                         map_trafo_H_1overx recursion;
1639                                         ex buffer = recursion(H(newparameter, arg).hold());
1640                                         if (is_a<add>(buffer)) {
1641                                                 for (int i=0; i<buffer.nops(); i++) {
1642                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
1643                                                 }
1644                                         } else {
1645                                                 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
1646                                         }
1647                                         return res;
1648
1649                                 } else {
1650
1651                                         // leading one
1652                                         map_trafo_H_1overx recursion;
1653                                         map_trafo_H_mult unify;
1654                                         ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1655                                         int firstzero = 0;
1656                                         while (parameter.op(firstzero) == 1) {
1657                                                 firstzero++;
1658                                         }
1659                                         for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1660                                                 lst newparameter;
1661                                                 int j=0;
1662                                                 for (; j<=i; j++) {
1663                                                         newparameter.append(parameter[j+1]);
1664                                                 }
1665                                                 newparameter.append(1);
1666                                                 for (; j<parameter.nops()-1; j++) {
1667                                                         newparameter.append(parameter[j+1]);
1668                                                 }
1669                                                 res -= H(newparameter, arg).hold();
1670                                         }
1671                                         res = recursion(res).expand() / firstzero;
1672                                         return unify(res);
1673
1674                                 }
1675
1676                         }
1677                 }
1678                 return e;
1679         }
1680 };
1681
1682
1683 // do x -> (1-x)/(1+x) transformation
1684 struct map_trafo_H_1mxt1px : public map_function
1685 {
1686         ex operator()(const ex& e)
1687         {
1688                 if (is_a<add>(e) || is_a<mul>(e)) {
1689                         return e.map(*this);
1690                 }
1691
1692                 if (is_a<function>(e)) {
1693                         std::string name = ex_to<function>(e).get_name();
1694                         if (name == "H") {
1695
1696                                 lst parameter = ex_to<lst>(e.op(0));
1697                                 ex arg = e.op(1);
1698
1699                                 // special cases if all parameters are either 0, 1 or -1
1700                                 bool allthesame = true;
1701                                 if (parameter.op(0) == 0) {
1702                                         for (int i=1; i<parameter.nops(); i++) {
1703                                                 if (parameter.op(i) != 0) {
1704                                                         allthesame = false;
1705                                                         break;
1706                                                 }
1707                                         }
1708                                         if (allthesame) {
1709                                                 map_trafo_H_mult unify;
1710                                                 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1711                                                        / factorial(parameter.nops())).expand());
1712                                         }
1713                                 } else if (parameter.op(0) == -1) {
1714                                         for (int i=1; i<parameter.nops(); i++) {
1715                                                 if (parameter.op(i) != -1) {
1716                                                         allthesame = false;
1717                                                         break;
1718                                                 }
1719                                         }
1720                                         if (allthesame) {
1721                                                 map_trafo_H_mult unify;
1722                                                 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1723                                                        / factorial(parameter.nops())).expand());
1724                                         }
1725                                 } else {
1726                                         for (int i=1; i<parameter.nops(); i++) {
1727                                                 if (parameter.op(i) != 1) {
1728                                                         allthesame = false;
1729                                                         break;
1730                                                 }
1731                                         }
1732                                         if (allthesame) {
1733                                                 map_trafo_H_mult unify;
1734                                                 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1735                                                        / factorial(parameter.nops())).expand());
1736                                         }
1737                                 }
1738
1739                                 lst newparameter = parameter;
1740                                 newparameter.remove_first();
1741
1742                                 if (parameter.op(0) == 0) {
1743
1744                                         // leading zero
1745                                         ex res = convert_H_to_zeta(parameter);
1746                                         map_trafo_H_1mxt1px recursion;
1747                                         ex buffer = recursion(H(newparameter, arg).hold());
1748                                         if (is_a<add>(buffer)) {
1749                                                 for (int i=0; i<buffer.nops(); i++) {
1750                                                         res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1751                                                 }
1752                                         } else {
1753                                                 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1754                                         }
1755                                         return res;
1756
1757                                 } else if (parameter.op(0) == -1) {
1758
1759                                         // leading negative one
1760                                         ex res = convert_H_to_zeta(parameter);
1761                                         map_trafo_H_1mxt1px recursion;
1762                                         ex buffer = recursion(H(newparameter, arg).hold());
1763                                         if (is_a<add>(buffer)) {
1764                                                 for (int i=0; i<buffer.nops(); i++) {
1765                                                         res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1766                                                 }
1767                                         } else {
1768                                                 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1769                                         }
1770                                         return res;
1771
1772                                 } else {
1773
1774                                         // leading one
1775                                         map_trafo_H_1mxt1px recursion;
1776                                         map_trafo_H_mult unify;
1777                                         ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1778                                         int firstzero = 0;
1779                                         while (parameter.op(firstzero) == 1) {
1780                                                 firstzero++;
1781                                         }
1782                                         for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1783                                                 lst newparameter;
1784                                                 int j=0;
1785                                                 for (; j<=i; j++) {
1786                                                         newparameter.append(parameter[j+1]);
1787                                                 }
1788                                                 newparameter.append(1);
1789                                                 for (; j<parameter.nops()-1; j++) {
1790                                                         newparameter.append(parameter[j+1]);
1791                                                 }
1792                                                 res -= H(newparameter, arg).hold();
1793                                         }
1794                                         res = recursion(res).expand() / firstzero;
1795                                         return unify(res);
1796
1797                                 }
1798
1799                         }
1800                 }
1801                 return e;
1802         }
1803 };
1804
1805
1806 // do the actual summation.
1807 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1808 {
1809         const int j = m.size();
1810
1811         std::vector<cln::cl_N> t(j);
1812
1813         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1814         cln::cl_N factor = cln::expt(x, j) * one;
1815         cln::cl_N t0buf;
1816         int q = 0;
1817         do {
1818                 t0buf = t[0];
1819                 q++;
1820                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1821                 for (int k=j-2; k>=1; k--) {
1822                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1823                 }
1824                 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1825                 factor = factor * x;
1826         } while (t[0] != t0buf);
1827
1828         return t[0];
1829 }
1830
1831
1832 } // end of anonymous namespace
1833
1834
1835 //////////////////////////////////////////////////////////////////////
1836 //
1837 // Harmonic polylogarithm  H(m,x)
1838 //
1839 // GiNaC function
1840 //
1841 //////////////////////////////////////////////////////////////////////
1842
1843
1844 static ex H_evalf(const ex& x1, const ex& x2)
1845 {
1846         if (is_a<lst>(x1) && is_a<numeric>(x2)) {
1847                 for (int i=0; i<x1.nops(); i++) {
1848                         if (!x1.op(i).info(info_flags::integer)) {
1849                                 return H(x1,x2).hold();
1850                         }
1851                 }
1852                 if (x1.nops() < 1) {
1853                         return H(x1,x2).hold();
1854                 }
1855
1856                 cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
1857                 
1858                 const lst& morg = ex_to<lst>(x1);
1859                 // remove trailing zeros ...
1860                 if (*(--morg.end()) == 0) {
1861                         symbol xtemp("xtemp");
1862                         map_trafo_H_reduce_trailing_zeros filter;
1863                         return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
1864                 }
1865                 // ... and expand parameter notation
1866                 lst m;
1867                 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
1868                         if (*it > 1) {
1869                                 for (ex count=*it-1; count > 0; count--) {
1870                                         m.append(0);
1871                                 }
1872                                 m.append(1);
1873                         } else if (*it < -1) {
1874                                 for (ex count=*it+1; count < 0; count++) {
1875                                         m.append(0);
1876                                 }
1877                                 m.append(-1);
1878                         } else {
1879                                 m.append(*it);
1880                         }
1881                 }
1882
1883                 // since the transformations produce a lot of terms, they are only efficient for
1884                 // argument near one.
1885                 // no transformation needed -> do summation
1886                 if (cln::abs(x) < 0.95) {
1887                         lst m_lst;
1888                         lst s_lst;
1889                         ex pf;
1890                         if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
1891                                 // negative parameters -> s_lst is filled
1892                                 std::vector<int> m_int;
1893                                 std::vector<cln::cl_N> x_cln;
1894                                 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin(); 
1895                                      it_int != m_lst.end(); it_int++, it_cln++) {
1896                                         m_int.push_back(ex_to<numeric>(*it_int).to_int());
1897                                         x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
1898                                 }
1899                                 x_cln.front() = x_cln.front() * x;
1900                                 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
1901                         } else {
1902                                 // only positive parameters
1903                                 //TODO
1904                                 if (m_lst.nops() == 1) {
1905                                         return Li(m_lst.op(0), x2).evalf();
1906                                 }
1907                                 std::vector<int> m_int;
1908                                 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
1909                                         m_int.push_back(ex_to<numeric>(*it).to_int());
1910                                 }
1911                                 return numeric(H_do_sum(m_int, x));
1912                         }
1913                 }
1914
1915                 ex res = 1;     
1916                 
1917                 // ensure that the realpart of the argument is positive
1918                 if (cln::realpart(x) < 0) {
1919                         x = -x;
1920                         for (int i=0; i<m.nops(); i++) {
1921                                 if (m.op(i) != 0) {
1922                                         m.let_op(i) = -m.op(i);
1923                                         res *= -1;
1924                                 }
1925                         }
1926                 }
1927
1928                 // choose transformations
1929                 symbol xtemp("xtemp");
1930                 if (cln::abs(x-1) < 1.4142) {
1931                         // x -> (1-x)/(1+x)
1932                         map_trafo_H_1mxt1px trafo;
1933                         res *= trafo(H(m, xtemp));
1934                 } else {
1935                         // x -> 1/x
1936                         map_trafo_H_1overx trafo;
1937                         res *= trafo(H(m, xtemp));
1938                         if (cln::imagpart(x) <= 0) {
1939                                 res = res.subs(H_polesign == -I*Pi);
1940                         } else {
1941                                 res = res.subs(H_polesign == I*Pi);
1942                         }
1943                 }
1944
1945                 // simplify result
1946 // TODO
1947 //              map_trafo_H_convert converter;
1948 //              res = converter(res).expand();
1949 //              lst ll;
1950 //              res.find(H(wild(1),wild(2)), ll);
1951 //              res.find(zeta(wild(1)), ll);
1952 //              res.find(zeta(wild(1),wild(2)), ll);
1953 //              res = res.collect(ll);
1954
1955                 return res.subs(xtemp == numeric(x)).evalf();
1956         }
1957
1958         return H(x1,x2).hold();
1959 }
1960
1961
1962 static ex H_eval(const ex& m_, const ex& x)
1963 {
1964         lst m;
1965         if (is_a<lst>(m_)) {
1966                 m = ex_to<lst>(m_);
1967         } else {
1968                 m = lst(m_);
1969         }
1970         if (m.nops() == 0) {
1971                 return _ex1;
1972         }
1973         ex pos1;
1974         ex pos2;
1975         ex n;
1976         ex p;
1977         int step = 0;
1978         if (*m.begin() > _ex1) {
1979                 step++;
1980                 pos1 = _ex0;
1981                 pos2 = _ex1;
1982                 n = *m.begin()-1;
1983                 p = _ex1;
1984         } else if (*m.begin() < _ex_1) {
1985                 step++;
1986                 pos1 = _ex0;
1987                 pos2 = _ex_1;
1988                 n = -*m.begin()-1;
1989                 p = _ex1;
1990         } else if (*m.begin() == _ex0) {
1991                 pos1 = _ex0;
1992                 n = _ex1;
1993         } else {
1994                 pos1 = *m.begin();
1995                 p = _ex1;
1996         }
1997         for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
1998                 if ((*it).info(info_flags::integer)) {
1999                         if (step == 0) {
2000                                 if (*it > _ex1) {
2001                                         if (pos1 == _ex0) {
2002                                                 step = 1;
2003                                                 pos2 = _ex1;
2004                                                 n += *it-1;
2005                                                 p = _ex1;
2006                                         } else {
2007                                                 step = 2;
2008                                         }
2009                                 } else if (*it < _ex_1) {
2010                                         if (pos1 == _ex0) {
2011                                                 step = 1;
2012                                                 pos2 = _ex_1;
2013                                                 n += -*it-1;
2014                                                 p = _ex1;
2015                                         } else {
2016                                                 step = 2;
2017                                         }
2018                                 } else {
2019                                         if (*it != pos1) {
2020                                                 step = 1;
2021                                                 pos2 = *it;
2022                                         }
2023                                         if (*it == _ex0) {
2024                                                 n++;
2025                                         } else {
2026                                                 p++;
2027                                         }
2028                                 }
2029                         } else if (step == 1) {
2030                                 if (*it != pos2) {
2031                                         step = 2;
2032                                 } else {
2033                                         if (*it == _ex0) {
2034                                                 n++;
2035                                         } else {
2036                                                 p++;
2037                                         }
2038                                 }
2039                         }
2040                 } else {
2041                         // if some m_i is not an integer
2042                         return H(m_, x).hold();
2043                 }
2044         }
2045         if ((x == _ex1) && (*(--m.end()) != _ex0)) {
2046                 return convert_H_to_zeta(m);
2047         }
2048         if (step == 0) {
2049                 if (pos1 == _ex0) {
2050                         // all zero
2051                         if (x == _ex0) {
2052                                 return H(m_, x).hold();
2053                         }
2054                         return pow(log(x), m.nops()) / factorial(m.nops());
2055                 } else {
2056                         // all (minus) one
2057                         return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
2058                 }
2059         } else if ((step == 1) && (pos1 == _ex0)){
2060                 // convertible to S
2061                 if (pos2 == _ex1) {
2062                         return S(n, p, x);
2063                 } else {
2064                         return pow(-1, p) * S(n, p, -x);
2065                 }
2066         }
2067         if (x == _ex0) {
2068                 return _ex0;
2069         }
2070         if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2071                 return H(m_, x).evalf();
2072         }
2073         return H(m_, x).hold();
2074 }
2075
2076
2077 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
2078 {
2079         epvector seq;
2080         seq.push_back(expair(H(m, x), 0));
2081         return pseries(rel, seq);
2082 }
2083
2084
2085 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
2086 {
2087         GINAC_ASSERT(deriv_param < 2);
2088         if (deriv_param == 0) {
2089                 return _ex0;
2090         }
2091         lst m;
2092         if (is_a<lst>(m_)) {
2093                 m = ex_to<lst>(m_);
2094         } else {
2095                 m = lst(m_);
2096         }
2097         ex mb = *m.begin();
2098         if (mb > _ex1) {
2099                 m[0]--;
2100                 return H(m, x) / x;
2101         }
2102         if (mb < _ex_1) {
2103                 m[0]++;
2104                 return H(m, x) / x;
2105         }
2106         m.remove_first();
2107         if (mb == _ex1) {
2108                 return 1/(1-x) * H(m, x);
2109         } else if (mb == _ex_1) {
2110                 return 1/(1+x) * H(m, x);
2111         } else {
2112                 return H(m, x) / x;
2113         }
2114 }
2115
2116
2117 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
2118 {
2119         lst m;
2120         if (is_a<lst>(m_)) {
2121                 m = ex_to<lst>(m_);
2122         } else {
2123                 m = lst(m_);
2124         }
2125         c.s << "\\mbox{H}_{";
2126         lst::const_iterator itm = m.begin();
2127         (*itm).print(c);
2128         itm++;
2129         for (; itm != m.end(); itm++) {
2130                 c.s << ",";
2131                 (*itm).print(c);
2132         }
2133         c.s << "}(";
2134         x.print(c);
2135         c.s << ")";
2136 }
2137
2138
2139 REGISTER_FUNCTION(H,
2140                   evalf_func(H_evalf).
2141                   eval_func(H_eval).
2142                   series_func(H_series).
2143                   derivative_func(H_deriv).
2144                   print_func<print_latex>(H_print_latex).
2145                   do_not_evalf_params());
2146
2147
2148 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
2149 ex convert_H_to_Li(const ex& m, const ex& x)
2150 {
2151         map_trafo_H_reduce_trailing_zeros filter;
2152         map_trafo_H_convert_to_Li filter2;
2153         if (is_a<lst>(m)) {
2154                 return filter2(filter(H(m, x).hold()));
2155         } else {
2156                 return filter2(filter(H(lst(m), x).hold()));
2157         }
2158 }
2159
2160
2161 //////////////////////////////////////////////////////////////////////
2162 //
2163 // Multiple zeta values  zeta(x) and zeta(x,s)
2164 //
2165 // helper functions
2166 //
2167 //////////////////////////////////////////////////////////////////////
2168
2169
2170 // anonymous namespace for helper functions
2171 namespace {
2172
2173
2174 // parameters and data for [Cra] algorithm
2175 const cln::cl_N lambda = cln::cl_N("319/320");
2176 int L1;
2177 int L2;
2178 std::vector<std::vector<cln::cl_N> > f_kj;
2179 std::vector<cln::cl_N> crB;
2180 std::vector<std::vector<cln::cl_N> > crG;
2181 std::vector<cln::cl_N> crX;
2182
2183
2184 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
2185 {
2186         const int size = a.size();
2187         for (int n=0; n<size; n++) {
2188                 c[n] = 0;
2189                 for (int m=0; m<=n; m++) {
2190                         c[n] = c[n] + a[m]*b[n-m];
2191                 }
2192         }
2193 }
2194
2195
2196 // [Cra] section 4
2197 void initcX(const std::vector<int>& s)
2198 {
2199         const int k = s.size();
2200
2201         crX.clear();
2202         crG.clear();
2203         crB.clear();
2204
2205         for (int i=0; i<=L2; i++) {
2206                 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
2207         }
2208
2209         int Sm = 0;
2210         int Smp1 = 0;
2211         for (int m=0; m<k-1; m++) {
2212                 std::vector<cln::cl_N> crGbuf;
2213                 Sm = Sm + s[m];
2214                 Smp1 = Sm + s[m+1];
2215                 for (int i=0; i<=L2; i++) {
2216                         crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
2217                 }
2218                 crG.push_back(crGbuf);
2219         }
2220
2221         crX = crB;
2222
2223         for (int m=0; m<k-1; m++) {
2224                 std::vector<cln::cl_N> Xbuf;
2225                 for (int i=0; i<=L2; i++) {
2226                         Xbuf.push_back(crX[i] * crG[m][i]);
2227                 }
2228                 halfcyclic_convolute(Xbuf, crB, crX);
2229         }
2230 }
2231
2232
2233 // [Cra] section 4
2234 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
2235 {
2236         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2237         cln::cl_N factor = cln::expt(lambda, Sqk);
2238         cln::cl_N res = factor / Sqk * crX[0] * one;
2239         cln::cl_N resbuf;
2240         int N = 0;
2241         do {
2242                 resbuf = res;
2243                 factor = factor * lambda;
2244                 N++;
2245                 res = res + crX[N] * factor / (N+Sqk);
2246         } while ((res != resbuf) || cln::zerop(crX[N]));
2247         return res;
2248 }
2249
2250
2251 // [Cra] section 4
2252 void calc_f(int maxr)
2253 {
2254         f_kj.clear();
2255         f_kj.resize(L1);
2256         
2257         cln::cl_N t0, t1, t2, t3, t4;
2258         int i, j, k;
2259         std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
2260         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2261         
2262         t0 = cln::exp(-lambda);
2263         t2 = 1;
2264         for (k=1; k<=L1; k++) {
2265                 t1 = k * lambda;
2266                 t2 = t0 * t2;
2267                 for (j=1; j<=maxr; j++) {
2268                         t3 = 1;
2269                         t4 = 1;
2270                         for (i=2; i<=j; i++) {
2271                                 t4 = t4 * (j-i+1);
2272                                 t3 = t1 * t3 + t4;
2273                         }
2274                         (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
2275                 }
2276                 it++;
2277         }
2278 }
2279
2280
2281 // [Cra] (3.1)
2282 cln::cl_N crandall_Z(const std::vector<int>& s)
2283 {
2284         const int j = s.size();
2285
2286         if (j == 1) {   
2287                 cln::cl_N t0;
2288                 cln::cl_N t0buf;
2289                 int q = 0;
2290                 do {
2291                         t0buf = t0;
2292                         q++;
2293                         t0 = t0 + f_kj[q+j-2][s[0]-1];
2294                 } while (t0 != t0buf);
2295                 
2296                 return t0 / cln::factorial(s[0]-1);
2297         }
2298
2299         std::vector<cln::cl_N> t(j);
2300
2301         cln::cl_N t0buf;
2302         int q = 0;
2303         do {
2304                 t0buf = t[0];
2305                 q++;
2306                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
2307                 for (int k=j-2; k>=1; k--) {
2308                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
2309                 }
2310                 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
2311         } while (t[0] != t0buf);
2312         
2313         return t[0] / cln::factorial(s[0]-1);
2314 }
2315
2316
2317 // [Cra] (2.4)
2318 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
2319 {
2320         std::vector<int> r = s;
2321         const int j = r.size();
2322
2323         // decide on maximal size of f_kj for crandall_Z
2324         if (Digits < 50) {
2325                 L1 = 150;
2326         } else {
2327                 L1 = Digits * 3 + j*2;
2328         }
2329
2330         // decide on maximal size of crX for crandall_Y
2331         if (Digits < 38) {
2332                 L2 = 63;
2333         } else if (Digits < 86) {
2334                 L2 = 127;
2335         } else if (Digits < 192) {
2336                 L2 = 255;
2337         } else if (Digits < 394) {
2338                 L2 = 511;
2339         } else if (Digits < 808) {
2340                 L2 = 1023;
2341         } else {
2342                 L2 = 2047;
2343         }
2344
2345         cln::cl_N res;
2346
2347         int maxr = 0;
2348         int S = 0;
2349         for (int i=0; i<j; i++) {
2350                 S += r[i];
2351                 if (r[i] > maxr) {
2352                         maxr = r[i];
2353                 }
2354         }
2355
2356         calc_f(maxr);
2357
2358         const cln::cl_N r0factorial = cln::factorial(r[0]-1);
2359
2360         std::vector<int> rz;
2361         int skp1buf;
2362         int Srun = S;
2363         for (int k=r.size()-1; k>0; k--) {
2364
2365                 rz.insert(rz.begin(), r.back());
2366                 skp1buf = rz.front();
2367                 Srun -= skp1buf;
2368                 r.pop_back();
2369
2370                 initcX(r);
2371                 
2372                 for (int q=0; q<skp1buf; q++) {
2373                         
2374                         cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2375                         cln::cl_N pp2 = crandall_Z(rz);
2376
2377                         rz.front()--;
2378                         
2379                         if (q & 1) {
2380                                 res = res - pp1 * pp2 / cln::factorial(q);
2381                         } else {
2382                                 res = res + pp1 * pp2 / cln::factorial(q);
2383                         }
2384                 }
2385                 rz.front() = skp1buf;
2386         }
2387         rz.insert(rz.begin(), r.back());
2388
2389         initcX(rz);
2390
2391         res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2392
2393         return res;
2394 }
2395
2396
2397 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2398 {
2399         const int j = r.size();
2400
2401         // buffer for subsums
2402         std::vector<cln::cl_N> t(j);
2403         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2404
2405         cln::cl_N t0buf;
2406         int q = 0;
2407         do {
2408                 t0buf = t[0];
2409                 q++;
2410                 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2411                 for (int k=j-2; k>=0; k--) {
2412                         t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2413                 }
2414         } while (t[0] != t0buf);
2415
2416         return t[0];
2417 }
2418
2419
2420 // does Hoelder convolution. see [BBB] (7.0)
2421 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2422 {
2423         // prepare parameters
2424         // holds Li arguments in [BBB] notation
2425         std::vector<int> s = s_;
2426         std::vector<int> m_p = m_;
2427         std::vector<int> m_q;
2428         // holds Li arguments in nested sums notation
2429         std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2430         s_p[0] = s_p[0] * cln::cl_N("1/2");
2431         // convert notations
2432         int sig = 1;
2433         for (int i=0; i<s_.size(); i++) {
2434                 if (s_[i] < 0) {
2435                         sig = -sig;
2436                         s_p[i] = -s_p[i];
2437                 }
2438                 s[i] = sig * std::abs(s[i]);
2439         }
2440         std::vector<cln::cl_N> s_q;
2441         cln::cl_N signum = 1;
2442
2443         // first term
2444         cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2445
2446         // middle terms
2447         do {
2448
2449                 // change parameters
2450                 if (s.front() > 0) {
2451                         if (m_p.front() == 1) {
2452                                 m_p.erase(m_p.begin());
2453                                 s_p.erase(s_p.begin());
2454                                 if (s_p.size() > 0) {
2455                                         s_p.front() = s_p.front() * cln::cl_N("1/2");
2456                                 }
2457                                 s.erase(s.begin());
2458                                 m_q.front()++;
2459                         } else {
2460                                 m_p.front()--;
2461                                 m_q.insert(m_q.begin(), 1);
2462                                 if (s_q.size() > 0) {
2463                                         s_q.front() = s_q.front() * 2;
2464                                 }
2465                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2466                         }
2467                 } else {
2468                         if (m_p.front() == 1) {
2469                                 m_p.erase(m_p.begin());
2470                                 cln::cl_N spbuf = s_p.front();
2471                                 s_p.erase(s_p.begin());
2472                                 if (s_p.size() > 0) {
2473                                         s_p.front() = s_p.front() * spbuf;
2474                                 }
2475                                 s.erase(s.begin());
2476                                 m_q.insert(m_q.begin(), 1);
2477                                 if (s_q.size() > 0) {
2478                                         s_q.front() = s_q.front() * 4;
2479                                 }
2480                                 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2481                                 signum = -signum;
2482                         } else {
2483                                 m_p.front()--;
2484                                 m_q.insert(m_q.begin(), 1);
2485                                 if (s_q.size() > 0) {
2486                                         s_q.front() = s_q.front() * 2;
2487                                 }
2488                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2489                         }
2490                 }
2491
2492                 // exiting the loop
2493                 if (m_p.size() == 0) break;
2494
2495                 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2496
2497         } while (true);
2498
2499         // last term
2500         res = res + signum * multipleLi_do_sum(m_q, s_q);
2501
2502         return res;
2503 }
2504
2505
2506 } // end of anonymous namespace
2507
2508
2509 //////////////////////////////////////////////////////////////////////
2510 //
2511 // Multiple zeta values  zeta(x)
2512 //
2513 // GiNaC function
2514 //
2515 //////////////////////////////////////////////////////////////////////
2516
2517
2518 static ex zeta1_evalf(const ex& x)
2519 {
2520         if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2521
2522                 // multiple zeta value
2523                 const int count = x.nops();
2524                 const lst& xlst = ex_to<lst>(x);
2525                 std::vector<int> r(count);
2526
2527                 // check parameters and convert them
2528                 lst::const_iterator it1 = xlst.begin();
2529                 std::vector<int>::iterator it2 = r.begin();
2530                 do {
2531                         if (!(*it1).info(info_flags::posint)) {
2532                                 return zeta(x).hold();
2533                         }
2534                         *it2 = ex_to<numeric>(*it1).to_int();
2535                         it1++;
2536                         it2++;
2537                 } while (it2 != r.end());
2538
2539                 // check for divergence
2540                 if (r[0] == 1) {
2541                         return zeta(x).hold();
2542                 }
2543
2544                 // decide on summation algorithm
2545                 // this is still a bit clumsy
2546                 int limit = (Digits>17) ? 10 : 6;
2547                 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2548                         return numeric(zeta_do_sum_Crandall(r));
2549                 } else {
2550                         return numeric(zeta_do_sum_simple(r));
2551                 }
2552         }
2553
2554         // single zeta value
2555         if (is_exactly_a<numeric>(x) && (x != 1)) {
2556                 try {
2557                         return zeta(ex_to<numeric>(x));
2558                 } catch (const dunno &e) { }
2559         }
2560
2561         return zeta(x).hold();
2562 }
2563
2564
2565 static ex zeta1_eval(const ex& m)
2566 {
2567         if (is_exactly_a<lst>(m)) {
2568                 if (m.nops() == 1) {
2569                         return zeta(m.op(0));
2570                 }
2571                 return zeta(m).hold();
2572         }
2573
2574         if (m.info(info_flags::numeric)) {
2575                 const numeric& y = ex_to<numeric>(m);
2576                 // trap integer arguments:
2577                 if (y.is_integer()) {
2578                         if (y.is_zero()) {
2579                                 return _ex_1_2;
2580                         }
2581                         if (y.is_equal(_num1)) {
2582                                 return zeta(m).hold();
2583                         }
2584                         if (y.info(info_flags::posint)) {
2585                                 if (y.info(info_flags::odd)) {
2586                                         return zeta(m).hold();
2587                                 } else {
2588                                         return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2589                                 }
2590                         } else {
2591                                 if (y.info(info_flags::odd)) {
2592                                         return -bernoulli(_num1-y) / (_num1-y);
2593                                 } else {
2594                                         return _ex0;
2595                                 }
2596                         }
2597                 }
2598                 // zeta(float)
2599                 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
2600                         return zeta1_evalf(m);
2601                 }
2602         }
2603         return zeta(m).hold();
2604 }
2605
2606
2607 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
2608 {
2609         GINAC_ASSERT(deriv_param==0);
2610
2611         if (is_exactly_a<lst>(m)) {
2612                 return _ex0;
2613         } else {
2614                 return zetaderiv(_ex1, m);
2615         }
2616 }
2617
2618
2619 static void zeta1_print_latex(const ex& m_, const print_context& c)
2620 {
2621         c.s << "\\zeta(";
2622         if (is_a<lst>(m_)) {
2623                 const lst& m = ex_to<lst>(m_);
2624                 lst::const_iterator it = m.begin();
2625                 (*it).print(c);
2626                 it++;
2627                 for (; it != m.end(); it++) {
2628                         c.s << ",";
2629                         (*it).print(c);
2630                 }
2631         } else {
2632                 m_.print(c);
2633         }
2634         c.s << ")";
2635 }
2636
2637
2638 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta").
2639                                 evalf_func(zeta1_evalf).
2640                                 eval_func(zeta1_eval).
2641                                 derivative_func(zeta1_deriv).
2642                                 print_func<print_latex>(zeta1_print_latex).
2643                                 do_not_evalf_params().
2644                                 overloaded(2));
2645
2646
2647 //////////////////////////////////////////////////////////////////////
2648 //
2649 // Alternating Euler sum  zeta(x,s)
2650 //
2651 // GiNaC function
2652 //
2653 //////////////////////////////////////////////////////////////////////
2654
2655
2656 static ex zeta2_evalf(const ex& x, const ex& s)
2657 {
2658         if (is_exactly_a<lst>(x)) {
2659
2660                 // alternating Euler sum
2661                 const int count = x.nops();
2662                 const lst& xlst = ex_to<lst>(x);
2663                 const lst& slst = ex_to<lst>(s);
2664                 std::vector<int> xi(count);
2665                 std::vector<int> si(count);
2666
2667                 // check parameters and convert them
2668                 lst::const_iterator it_xread = xlst.begin();
2669                 lst::const_iterator it_sread = slst.begin();
2670                 std::vector<int>::iterator it_xwrite = xi.begin();
2671                 std::vector<int>::iterator it_swrite = si.begin();
2672                 do {
2673                         if (!(*it_xread).info(info_flags::posint)) {
2674                                 return zeta(x, s).hold();
2675                         }
2676                         *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2677                         if (*it_sread > 0) {
2678                                 *it_swrite = 1;
2679                         } else {
2680                                 *it_swrite = -1;
2681                         }
2682                         it_xread++;
2683                         it_sread++;
2684                         it_xwrite++;
2685                         it_swrite++;
2686                 } while (it_xwrite != xi.end());
2687
2688                 // check for divergence
2689                 if ((xi[0] == 1) && (si[0] == 1)) {
2690                         return zeta(x, s).hold();
2691                 }
2692
2693                 // use Hoelder convolution
2694                 return numeric(zeta_do_Hoelder_convolution(xi, si));
2695         }
2696
2697         return zeta(x, s).hold();
2698 }
2699
2700
2701 static ex zeta2_eval(const ex& m, const ex& s_)
2702 {
2703         if (is_exactly_a<lst>(s_)) {
2704                 const lst& s = ex_to<lst>(s_);
2705                 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
2706                         if ((*it).info(info_flags::positive)) {
2707                                 continue;
2708                         }
2709                         return zeta(m, s_).hold();
2710                 }
2711                 return zeta(m);
2712         } else if (s_.info(info_flags::positive)) {
2713                 return zeta(m);
2714         }
2715
2716         return zeta(m, s_).hold();
2717 }
2718
2719
2720 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
2721 {
2722         GINAC_ASSERT(deriv_param==0);
2723
2724         if (is_exactly_a<lst>(m)) {
2725                 return _ex0;
2726         } else {
2727                 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
2728                         return zetaderiv(_ex1, m);
2729                 }
2730                 return _ex0;
2731         }
2732 }
2733
2734
2735 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
2736 {
2737         lst m;
2738         if (is_a<lst>(m_)) {
2739                 m = ex_to<lst>(m_);
2740         } else {
2741                 m = lst(m_);
2742         }
2743         lst s;
2744         if (is_a<lst>(s_)) {
2745                 s = ex_to<lst>(s_);
2746         } else {
2747                 s = lst(s_);
2748         }
2749         c.s << "\\zeta(";
2750         lst::const_iterator itm = m.begin();
2751         lst::const_iterator its = s.begin();
2752         if (*its < 0) {
2753                 c.s << "\\overline{";
2754                 (*itm).print(c);
2755                 c.s << "}";
2756         } else {
2757                 (*itm).print(c);
2758         }
2759         its++;
2760         itm++;
2761         for (; itm != m.end(); itm++, its++) {
2762                 c.s << ",";
2763                 if (*its < 0) {
2764                         c.s << "\\overline{";
2765                         (*itm).print(c);
2766                         c.s << "}";
2767                 } else {
2768                         (*itm).print(c);
2769                 }
2770         }
2771         c.s << ")";
2772 }
2773
2774
2775 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta").
2776                                 evalf_func(zeta2_evalf).
2777                                 eval_func(zeta2_eval).
2778                                 derivative_func(zeta2_deriv).
2779                                 print_func<print_latex>(zeta2_print_latex).
2780                                 do_not_evalf_params().
2781                                 overloaded(2));
2782
2783
2784 } // namespace GiNaC
2785