Algorithmic Methods for Special Functions by Computer Algebra

Papers and software produced in the project

• paper A generalization of Kummer's identity by R. Vidunas, Rocky Mountain J. Math. 32 (2002), 919-936; arXiv:math.CA/0005095.
  An accompanying Maple worksheet kummer.mws by R. Vidunas joint with T.H. Koornwinder, 2000.
• Maple procedure infhsum.mpl (version 4.29, 30 November 2005) for extension of Zeilberger's algorithm to nonterminating series (by R. Vidunas).
  There is an accompanying Maple worksheet infhsum.mws by R. Vidunas.
  The procedure and worksheets call W. Koepf's hsum9.mpl. This material has been tested in Maple 7, 8 and 9.
• paper Symbolic evaluation of coefficients in Airy-type asymptotic expansions by R. Vidunas and N.M. Temme, J. Math. Anal. Appl. 269 (2002), 317-331; arXiv:math.CA/0103184.
  By the same authors: an accompanying Maple package airypcf.mpl (version 3.0, 17 January 2001) and a Maple worksheet airypcf.mws demonstrating the usage of the package.
• paper Contiguous relations of hypergeometric series by R. Vidunas, J. Comput. Appl. Math. 153 (2003), 507-519; arXiv:math.CA/0109222 v4 corrects formulas (24) and (25) in that journal article.
  See also an accompanying Maple package contiguous2f1.mpl and a Maple worksheet contiguous2f1demo.mws illustrating the usage of this package,
• paper Parabolic cylinder functions: examples of error bounds for asymptotic expansions by N.M. Temme and R. Vidunas, Anal. Appl. (Singap.) 1 (2003), 265-288; arXiv:math.CA/0205045.
• paper Transformations of some Gauss hypergeometric functions by R. Vidunas, J. Comput. Appl. Math. 178 (2005), 473-487; arXiv:math.CA/0310436.
• preprint Uniform convergence of hypergeometric series by R. Vidunas, arXiv:math.CA/0603646 v2.

Related papers and software produced outside the project

• paper Expressions for values of the gamma function by R. Vidunas, Kyushu J. Math. 59 (2005), 267-283; arXiv:math.CA/0403510.
  See also an accompanying Maple package gammas120.mpl and a Mathematica package gammas120.m for automated simplification of expressions with gamma values at rational points with the denominator dividing 120.

Literature

• The book A=B by M. Petkovsek, H.S. Wilf and D. Zeilberger, publisher A.K. Peters, Wellesley, Massachusetts, 1996.
  On the web page accompanying this book you find links to:
  • Free download of the book
  • Errata sheet to the book
  • Programs, in particular EKHAD and qEKHAD (updated for Maple 6 in September 2000; EKHAD also updated for Maple 8)
  • Case Studies
• The book Hypergeometric Summation. An Algorithmic Approach to Summation and Special Function Identities by Wolfram Koepf, Vieweg Advanced Lectures in Mathematics, 1998.
  Links to Koepf's accompanying software:
  • hsum13.mpl
  • maple.hdb
  • qsum13.mpl
  • The article Algorithms for q-Hypergeometric Summation in Computer Algebra (J. Symbolic Comput. 28 (1999), 777-799, with H. Böing), which gives a description of the package qsum.mpl.

Further Software

• Computer Programs From the Book "A=B", and Related Programs (page by H. Wilf)
• Description of Software on symbolic summation by P. Paule's combinatorics group at RISC/Linz. This is mostly in Mathematica.
• The Mathematica packages HYP and HYPQ by Christian Krattenthaler
• The Maple implementations zeilb and qzeilb by Tom Koornwinder of the (q-)Zeilberger algorithm.
• Algorithms project at INRIA (B. Salvy, F. Chyzak and others). See the package gfun on generating functions.
• Marko Petkovsek, University of Ljubljana. See some Mathematica packages.

General links

• Computer algebra
• Special functions
• People in Symbolic Computation and Algebraic Combinatorics (Marko Petkovsek's list)