ginac/inifcns_gamma.cpp

   1 /** @file inifcns_gamma.cpp
   2  *
   3  *  Implementation of Gamma-function, Beta-function, Polygamma-functions, and
   4  *  some related stuff. */
   5
   6 /*
   7  *  GiNaC Copyright (C) 1999-2018 Johannes Gutenberg University Mainz, Germany
   8  *
   9  *  This program is free software; you can redistribute it and/or modify
  10  *  it under the terms of the GNU General Public License as published by
  11  *  the Free Software Foundation; either version 2 of the License, or
  12  *  (at your option) any later version.
  13  *
  14  *  This program is distributed in the hope that it will be useful,
  15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  17  *  GNU General Public License for more details.
  18  *
  19  *  You should have received a copy of the GNU General Public License
  20  *  along with this program; if not, write to the Free Software
  21  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  22  */
  23
  24 #include "inifcns.h"
  25 #include "constant.h"
  26 #include "pseries.h"
  27 #include "numeric.h"
  28 #include "power.h"
  29 #include "relational.h"
  30 #include "operators.h"
  31 #include "symbol.h"
  32 #include "symmetry.h"
  33 #include "utils.h"
  34
  35 #include <stdexcept>
  36 #include <vector>
  37
  38 namespace GiNaC {
  39
  40 //////////
  41 // Logarithm of Gamma function
  42 //////////
  43
  44 static ex lgamma_evalf(const ex & x)
  45 {
  46         if (is_exactly_a<numeric>(x)) {
  47                 try {
  48                         return lgamma(ex_to<numeric>(x));
  49                 } catch (const dunno &e) { }
  50         }
  51
  52         return lgamma(x).hold();
  53 }
  54
  55
  56 /** Evaluation of lgamma(x), the natural logarithm of the Gamma function.
  57  *  Handles integer arguments as a special case.
  58  *
  59  *  @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */
  60 static ex lgamma_eval(const ex & x)
  61 {
  62         if (x.info(info_flags::numeric)) {
  63                 // trap integer arguments:
  64                 if (x.info(info_flags::integer)) {
  65                         // lgamma(n) -> log((n-1)!) for postitive n
  66                         if (x.info(info_flags::posint))
  67                                 return log(factorial(x + _ex_1));
  68                         else
  69                                 throw (pole_error("lgamma_eval(): logarithmic pole",0));
  70                 }
  71                 if (!ex_to<numeric>(x).is_rational())
  72                         return lgamma(ex_to<numeric>(x));
  73         }
  74
  75         return lgamma(x).hold();
  76 }
  77
  78
  79 static ex lgamma_deriv(const ex & x, unsigned deriv_param)
  80 {
  81         GINAC_ASSERT(deriv_param==0);
  82
  83         // d/dx  lgamma(x) -> psi(x)
  84         return psi(x);
  85 }
  86
  87
  88 static ex lgamma_series(const ex & arg,
  89                         const relational & rel,
  90                         int order,
  91                         unsigned options)
  92 {
  93         // method:
  94         // Taylor series where there is no pole falls back to psi function
  95         // evaluation.
  96         // On a pole at -m we could use the recurrence relation
  97         //   lgamma(x) == lgamma(x+1)-log(x)
  98         // from which follows
  99         //   series(lgamma(x),x==-m,order) ==
 100         //   series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order);
 101         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 102         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 103                 throw do_taylor();  // caught by function::series()
 104         // if we got here we have to care for a simple pole of tgamma(-m):
 105         numeric m = -ex_to<numeric>(arg_pt);
 106         ex recur;
 107         for (numeric p = 0; p<=m; ++p)
 108                 recur += log(arg+p);
 109         return (lgamma(arg+m+_ex1)-recur).series(rel, order, options);
 110 }
 111
 112
 113 static ex lgamma_conjugate(const ex & x)
 114 {
 115         // conjugate(lgamma(x))==lgamma(conjugate(x)) unless on the branch cut
 116         // which runs along the negative real axis.
 117         if (x.info(info_flags::positive)) {
 118                 return lgamma(x);
 119         }
 120         if (is_exactly_a<numeric>(x) &&
 121             !x.imag_part().is_zero()) {
 122                 return lgamma(x.conjugate());
 123         }
 124         return conjugate_function(lgamma(x)).hold();
 125 }
 126
 127
 128 REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval).
 129                           evalf_func(lgamma_evalf).
 130                           derivative_func(lgamma_deriv).
 131                           series_func(lgamma_series).
 132                           conjugate_func(lgamma_conjugate).
 133                           latex_name("\\log \\Gamma"));
 134
 135
 136 //////////
 137 // true Gamma function
 138 //////////
 139
 140 static ex tgamma_evalf(const ex & x)
 141 {
 142         if (is_exactly_a<numeric>(x)) {
 143                 try {
 144                         return tgamma(ex_to<numeric>(x));
 145                 } catch (const dunno &e) { }
 146         }
 147
 148         return tgamma(x).hold();
 149 }
 150
 151
 152 /** Evaluation of tgamma(x), the true Gamma function.  Knows about integer
 153  *  arguments, half-integer arguments and that's it. Somebody ought to provide
 154  *  some good numerical evaluation some day...
 155  *
 156  *  @exception pole_error("tgamma_eval(): simple pole",0) */
 157 static ex tgamma_eval(const ex & x)
 158 {
 159         if (x.info(info_flags::numeric)) {
 160                 // trap integer arguments:
 161                 const numeric two_x = (*_num2_p)*ex_to<numeric>(x);
 162                 if (two_x.is_even()) {
 163                         // tgamma(n) -> (n-1)! for postitive n
 164                         if (two_x.is_positive()) {
 165                                 return factorial(ex_to<numeric>(x).sub(*_num1_p));
 166                         } else {
 167                                 throw (pole_error("tgamma_eval(): simple pole",1));
 168                         }
 169                 }
 170                 // trap half integer arguments:
 171                 if (two_x.is_integer()) {
 172                         // trap positive x==(n+1/2)
 173                         // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n)
 174                         if (two_x.is_positive()) {
 175                                 const numeric n = ex_to<numeric>(x).sub(*_num1_2_p);
 176                                 return (doublefactorial(n.mul(*_num2_p).sub(*_num1_p)).div(pow(*_num2_p,n))) * sqrt(Pi);
 177                         } else {
 178                                 // trap negative x==(-n+1/2)
 179                                 // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
 180                                 const numeric n = abs(ex_to<numeric>(x).sub(*_num1_2_p));
 181                                 return (pow(*_num_2_p, n).div(doublefactorial(n.mul(*_num2_p).sub(*_num1_p))))*sqrt(Pi);
 182                         }
 183                 }
 184                 if (!ex_to<numeric>(x).is_rational())
 185                         return tgamma(ex_to<numeric>(x));
 186         }
 187
 188         return tgamma(x).hold();
 189 }
 190
 191
 192 static ex tgamma_deriv(const ex & x, unsigned deriv_param)
 193 {
 194         GINAC_ASSERT(deriv_param==0);
 195
 196         // d/dx  tgamma(x) -> psi(x)*tgamma(x)
 197         return psi(x)*tgamma(x);
 198 }
 199
 200
 201 static ex tgamma_series(const ex & arg,
 202                         const relational & rel,
 203                         int order,
 204                         unsigned options)
 205 {
 206         // method:
 207         // Taylor series where there is no pole falls back to psi function
 208         // evaluation.
 209         // On a pole at -m use the recurrence relation
 210         //   tgamma(x) == tgamma(x+1) / x
 211         // from which follows
 212         //   series(tgamma(x),x==-m,order) ==
 213         //   series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order);
 214         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 215         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 216                 throw do_taylor();  // caught by function::series()
 217         // if we got here we have to care for a simple pole at -m:
 218         const numeric m = -ex_to<numeric>(arg_pt);
 219         ex ser_denom = _ex1;
 220         for (numeric p; p<=m; ++p)
 221                 ser_denom *= arg+p;
 222         return (tgamma(arg+m+_ex1)/ser_denom).series(rel, order, options);
 223 }
 224
 225
 226 static ex tgamma_conjugate(const ex & x)
 227 {
 228         // conjugate(tgamma(x))==tgamma(conjugate(x))
 229         return tgamma(x.conjugate());
 230 }
 231
 232
 233 REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval).
 234                           evalf_func(tgamma_evalf).
 235                           derivative_func(tgamma_deriv).
 236                           series_func(tgamma_series).
 237                           conjugate_func(tgamma_conjugate).
 238                           latex_name("\\Gamma"));
 239
 240
 241 //////////
 242 // beta-function
 243 //////////
 244
 245 static ex beta_evalf(const ex & x, const ex & y)
 246 {
 247         if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y)) {
 248                 try {
 249                         return exp(lgamma(ex_to<numeric>(x))+lgamma(ex_to<numeric>(y))-lgamma(ex_to<numeric>(x+y)));
 250                 } catch (const dunno &e) { }
 251         }
 252
 253         return beta(x,y).hold();
 254 }
 255
 256
 257 static ex beta_eval(const ex & x, const ex & y)
 258 {
 259         if (x.is_equal(_ex1))
 260                 return 1/y;
 261         if (y.is_equal(_ex1))
 262                 return 1/x;
 263         if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
 264                 // treat all problematic x and y that may not be passed into tgamma,
 265                 // because they would throw there although beta(x,y) is well-defined
 266                 // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
 267                 const numeric &nx = ex_to<numeric>(x);
 268                 const numeric &ny = ex_to<numeric>(y);
 269                 if (nx.is_real() && nx.is_integer() &&
 270                         ny.is_real() && ny.is_integer()) {
 271                         if (nx.is_negative()) {
 272                                 if (nx<=-ny)
 273                                         return pow(*_num_1_p, ny)*beta(1-x-y, y);
 274                                 else
 275                                         throw (pole_error("beta_eval(): simple pole",1));
 276                         }
 277                         if (ny.is_negative()) {
 278                                 if (ny<=-nx)
 279                                         return pow(*_num_1_p, nx)*beta(1-y-x, x);
 280                                 else
 281                                         throw (pole_error("beta_eval(): simple pole",1));
 282                         }
 283                         return tgamma(x)*tgamma(y)/tgamma(x+y);
 284                 }
 285                 // no problem in numerator, but denominator has pole:
 286                 if ((nx+ny).is_real() &&
 287                     (nx+ny).is_integer() &&
 288                    !(nx+ny).is_positive())
 289                          return _ex0;
 290                 if (!ex_to<numeric>(x).is_rational() || !ex_to<numeric>(x).is_rational())
 291                         return evalf(beta(x, y).hold());
 292         }
 293
 294         return beta(x,y).hold();
 295 }
 296
 297
 298 static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param)
 299 {
 300         GINAC_ASSERT(deriv_param<2);
 301         ex retval;
 302
 303         // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y)
 304         if (deriv_param==0)
 305                 retval = (psi(x)-psi(x+y))*beta(x,y);
 306         // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y)
 307         if (deriv_param==1)
 308                 retval = (psi(y)-psi(x+y))*beta(x,y);
 309         return retval;
 310 }
 311
 312
 313 static ex beta_series(const ex & arg1,
 314                       const ex & arg2,
 315                       const relational & rel,
 316                       int order,
 317                       unsigned options)
 318 {
 319         // method:
 320         // Taylor series where there is no pole of one of the tgamma functions
 321         // falls back to beta function evaluation.  Otherwise, fall back to
 322         // tgamma series directly.
 323         const ex arg1_pt = arg1.subs(rel, subs_options::no_pattern);
 324         const ex arg2_pt = arg2.subs(rel, subs_options::no_pattern);
 325         GINAC_ASSERT(is_a<symbol>(rel.lhs()));
 326         const symbol &s = ex_to<symbol>(rel.lhs());
 327         ex arg1_ser, arg2_ser, arg1arg2_ser;
 328         if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) &&
 329             (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive)))
 330                 throw do_taylor();  // caught by function::series()
 331         // trap the case where arg1 is on a pole:
 332         if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive))
 333                 arg1_ser = tgamma(arg1+s);
 334         else
 335                 arg1_ser = tgamma(arg1);
 336         // trap the case where arg2 is on a pole:
 337         if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive))
 338                 arg2_ser = tgamma(arg2+s);
 339         else
 340                 arg2_ser = tgamma(arg2);
 341         // trap the case where arg1+arg2 is on a pole:
 342         if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive))
 343                 arg1arg2_ser = tgamma(arg2+arg1+s);
 344         else
 345                 arg1arg2_ser = tgamma(arg2+arg1);
 346         // compose the result (expanding all the terms):
 347         return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand();
 348 }
 349
 350
 351 REGISTER_FUNCTION(beta, eval_func(beta_eval).
 352                         evalf_func(beta_evalf).
 353                         derivative_func(beta_deriv).
 354                         series_func(beta_series).
 355                         latex_name("\\mathrm{B}").
 356                         set_symmetry(sy_symm(0, 1)));
 357
 358
 359 //////////
 360 // Psi-function (aka digamma-function)
 361 //////////
 362
 363 static ex psi1_evalf(const ex & x)
 364 {
 365         if (is_exactly_a<numeric>(x)) {
 366                 try {
 367                         return psi(ex_to<numeric>(x));
 368                 } catch (const dunno &e) { }
 369         }
 370
 371         return psi(x).hold();
 372 }
 373
 374 /** Evaluation of digamma-function psi(x).
 375  *  Somebody ought to provide some good numerical evaluation some day... */
 376 static ex psi1_eval(const ex & x)
 377 {
 378         if (x.info(info_flags::numeric)) {
 379                 const numeric &nx = ex_to<numeric>(x);
 380                 if (nx.is_integer()) {
 381                         // integer case
 382                         if (nx.is_positive()) {
 383                                 // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler
 384                                 numeric rat = 0;
 385                                 for (numeric i(nx+(*_num_1_p)); i>0; --i)
 386                                         rat += i.inverse();
 387                                 return rat-Euler;
 388                         } else {
 389                                 // for non-positive integers there is a pole:
 390                                 throw (pole_error("psi_eval(): simple pole",1));
 391                         }
 392                 }
 393                 if (((*_num2_p)*nx).is_integer()) {
 394                         // half integer case
 395                         if (nx.is_positive()) {
 396                                 // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2)
 397                                 numeric rat = 0;
 398                                 for (numeric i = (nx+(*_num_1_p))*(*_num2_p); i>0; i-=(*_num2_p))
 399                                         rat += (*_num2_p)*i.inverse();
 400                                 return rat-Euler-_ex2*log(_ex2);
 401                         } else {
 402                                 // use the recurrence relation
 403                                 //   psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2)
 404                                 // to relate psi(-m-1/2) to psi(1/2):
 405                                 //   psi(-m-1/2) == psi(1/2) + r
 406                                 // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1))
 407                                 numeric recur = 0;
 408                                 for (numeric p = nx; p<0; ++p)
 409                                         recur -= pow(p, *_num_1_p);
 410                                 return recur+psi(_ex1_2);
 411                         }
 412                 }
 413                 //  psi1_evalf should be called here once it becomes available
 414         }
 415
 416         return psi(x).hold();
 417 }
 418
 419 static ex psi1_deriv(const ex & x, unsigned deriv_param)
 420 {
 421         GINAC_ASSERT(deriv_param==0);
 422
 423         // d/dx psi(x) -> psi(1,x)
 424         return psi(_ex1, x);
 425 }
 426
 427 static ex psi1_series(const ex & arg,
 428                       const relational & rel,
 429                       int order,
 430                       unsigned options)
 431 {
 432         // method:
 433         // Taylor series where there is no pole falls back to polygamma function
 434         // evaluation.
 435         // On a pole at -m use the recurrence relation
 436         //   psi(x) == psi(x+1) - 1/z
 437         // from which follows
 438         //   series(psi(x),x==-m,order) ==
 439         //   series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x==-m,order);
 440         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 441         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 442                 throw do_taylor();  // caught by function::series()
 443         // if we got here we have to care for a simple pole at -m:
 444         const numeric m = -ex_to<numeric>(arg_pt);
 445         ex recur;
 446         for (numeric p; p<=m; ++p)
 447                 recur += power(arg+p,_ex_1);
 448         return (psi(arg+m+_ex1)-recur).series(rel, order, options);
 449 }
 450
 451 unsigned psi1_SERIAL::serial =
 452         function::register_new(function_options("psi", 1).
 453                                eval_func(psi1_eval).
 454                                evalf_func(psi1_evalf).
 455                                derivative_func(psi1_deriv).
 456                                series_func(psi1_series).
 457                                latex_name("\\psi").
 458                                overloaded(2));
 459
 460 //////////
 461 // Psi-functions (aka polygamma-functions)  psi(0,x)==psi(x)
 462 //////////
 463
 464 static ex psi2_evalf(const ex & n, const ex & x)
 465 {
 466         if (is_exactly_a<numeric>(n) && is_exactly_a<numeric>(x)) {
 467                 try {
 468                         return psi(ex_to<numeric>(n),ex_to<numeric>(x));
 469                 } catch (const dunno &e) { }
 470         }
 471
 472         return psi(n,x).hold();
 473 }
 474
 475 /** Evaluation of polygamma-function psi(n,x).
 476  *  Somebody ought to provide some good numerical evaluation some day... */
 477 static ex psi2_eval(const ex & n, const ex & x)
 478 {
 479         // psi(0,x) -> psi(x)
 480         if (n.is_zero())
 481                 return psi(x);
 482         // psi(-1,x) -> log(tgamma(x))
 483         if (n.is_equal(_ex_1))
 484                 return log(tgamma(x));
 485         if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
 486                 x.info(info_flags::numeric)) {
 487                 const numeric &nn = ex_to<numeric>(n);
 488                 const numeric &nx = ex_to<numeric>(x);
 489                 if (nx.is_integer()) {
 490                         // integer case
 491                         if (nx.is_equal(*_num1_p))
 492                                 // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
 493                                 return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*zeta(ex(nn+(*_num1_p)));
 494                         if (nx.is_positive()) {
 495                                 // use the recurrence relation
 496                                 //   psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
 497                                 // to relate psi(n,m) to psi(n,1):
 498                                 //   psi(n,m) == psi(n,1) + r
 499                                 // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
 500                                 numeric recur = 0;
 501                                 for (numeric p = 1; p<nx; ++p)
 502                                         recur += pow(p, -nn+(*_num_1_p));
 503                                 recur *= factorial(nn)*pow((*_num_1_p), nn);
 504                                 return recur+psi(n,_ex1);
 505                         } else {
 506                                 // for non-positive integers there is a pole:
 507                                 throw (pole_error("psi2_eval(): pole",1));
 508                         }
 509                 }
 510                 if (((*_num2_p)*nx).is_integer()) {
 511                         // half integer case
 512                         if (nx.is_equal(*_num1_2_p))
 513                                 // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
 514                                 return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*(pow(*_num2_p,nn+(*_num1_p)) + (*_num_1_p))*zeta(ex(nn+(*_num1_p)));
 515                         if (nx.is_positive()) {
 516                                 const numeric m = nx - (*_num1_2_p);
 517                                 // use the multiplication formula
 518                                 //   psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
 519                                 // to revert to positive integer case
 520                                 return psi(n,(*_num2_p)*m)*pow((*_num2_p),nn+(*_num1_p))-psi(n,m);
 521                         } else {
 522                                 // use the recurrence relation
 523                                 //   psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
 524                                 // to relate psi(n,-m-1/2) to psi(n,1/2):
 525                                 //   psi(n,-m-1/2) == psi(n,1/2) + r
 526                                 // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
 527                                 numeric recur = 0;
 528                                 for (numeric p = nx; p<0; ++p)
 529                                         recur += pow(p, -nn+(*_num_1_p));
 530                                 recur *= factorial(nn)*pow(*_num_1_p, nn+(*_num_1_p));
 531                                 return recur+psi(n,_ex1_2);
 532                         }
 533                 }
 534                 //  psi2_evalf should be called here once it becomes available
 535         }
 536
 537         return psi(n, x).hold();
 538 }
 539
 540 static ex psi2_deriv(const ex & n, const ex & x, unsigned deriv_param)
 541 {
 542         GINAC_ASSERT(deriv_param<2);
 543
 544         if (deriv_param==0) {
 545                 // d/dn psi(n,x)
 546                 throw(std::logic_error("cannot diff psi(n,x) with respect to n"));
 547         }
 548         // d/dx psi(n,x) -> psi(n+1,x)
 549         return psi(n+_ex1, x);
 550 }
 551
 552 static ex psi2_series(const ex & n,
 553                       const ex & arg,
 554                       const relational & rel,
 555                       int order,
 556                       unsigned options)
 557 {
 558         // method:
 559         // Taylor series where there is no pole falls back to polygamma function
 560         // evaluation.
 561         // On a pole at -m use the recurrence relation
 562         //   psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1)
 563         // from which follows
 564         //   series(psi(x),x==-m,order) ==
 565         //   series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
 566         //                                      ... + (x+m)^(-n-1))),x==-m,order);
 567         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 568         if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive))
 569                 throw do_taylor();  // caught by function::series()
 570         // if we got here we have to care for a pole of order n+1 at -m:
 571         const numeric m = -ex_to<numeric>(arg_pt);
 572         ex recur;
 573         for (numeric p; p<=m; ++p)
 574                 recur += power(arg+p,-n+_ex_1);
 575         recur *= factorial(n)*power(_ex_1,n);
 576         return (psi(n, arg+m+_ex1)-recur).series(rel, order, options);
 577 }
 578
 579 unsigned psi2_SERIAL::serial =
 580         function::register_new(function_options("psi", 2).
 581                                eval_func(psi2_eval).
 582                                evalf_func(psi2_evalf).
 583                                derivative_func(psi2_deriv).
 584                                series_func(psi2_series).
 585                                latex_name("\\psi").
 586                                overloaded(2));
 587
 588
 589 } // namespace GiNaC