HolonomicFunctions

Short Description

The HolonomicFunctions package by Christoph Koutschan allows to deal with multivariate holonomic functions and sequences in an algorithmic fashion. For this purpose the package can compute annihilating ideals and execute closure properties (addition, multiplication, substitutions) for such functions. An annihilating ideal represents the set of linear differential equations, linear recurrences, q-difference equations, and mixed linear equations that a given function satisfies. Summation and integration of multivariate holonomic functions can be performed via creative telescoping. As subtasks, the following functionalities have been implemented in HolonomicFunctions: computations in Ore algebras (noncommutative polynomial arithmetic with mixed difference-differential operators), noncommutative Gröbner bases, and solving of coupled linear systems of differential or difference equations.

Registration and Legal Notices

The source code for this package is password protected. To get the password send an email to Peter Paule. It will be given for free to all researchers and non-commercial users.

A Note on Encoded Files

This package contains one or more Mathematica input files which are encoded. Those files cannot be read or modified directly as plain text, but can be loaded into Mathematica just like any normal input file (i.e., with <<"file" or Get["file"]). There is no need (and also no way) to decode them by using additional software or a special key.

If loading an encoded file causes a syntax error, open it with a text editor and remove any blank lines at the beginning (for some reason your Mac could have inserted them silently...).

The Package

The package is contained in the Mathematica input file

HolonomicFunctions.m (encoded)

and is accompanied by a large collection of examples to demonstrate its usage:

ExamplesV7.nb (for Mathematica version 7, last updated on August 9, 2011)
ExamplesV6.nb (for Mathematica version 6, last updated on August 9, 2011)
ExamplesV5.2.nb (for Mathematica version 5.2, last updated on August 9, 2011)

Right now you are using Version 1.6 released on April 12, 2012. This version is compatible with Mathematica versions from 5.2 to 8.0. Please report any bugs and comments to Christoph Koutschan.

Literature

The theoretical background of the algorithms implemented in HolonomicFunctions is described in

Advanced Applications of the Holonomic Systems Approach (PhD Thesis). RISC-Linz, Johannes Kepler University, September 2009. [pdf]
A Fast Approach to Creative Telescoping. Mathematics in Computer Science 4(2-3), pp. 259-266. 2010. [pdf]

The PhD thesis also contains a chapter about how to use the package.
All the commands that are contained in HolonomicFunctions are described in detail in the documentation

HolonomicFunctions (User's Guide). Christoph Koutschan. Technical report no. 10-01 in RISC Report Series, University of Linz, Austria. [pdf]

Some Applications

The package HolonomicFunctions has been applied in many different contexts, some of which are listed below.

From version 1.5.1 on, HolonomicFunctions provides the closure property twisting q-holonomic sequences by complex roots of unity via the command DFiniteQSubstitute; more details, examples, and applications in quantum topology (Kashaev invariant of twist and pretzel knots) are presented in the corresponding paper (see the above link).
In Advanced Computer Algebra for Determinants the package HolonomicFunctions was used to carry out Zeilberger's holonomic ansatz (and variations thereof) for determinant evaluations to solve three conjectures by George Andrews, Guoce Xin, and Christian Krattenthaler.
The proofs of some evaluations of Pfaffians have been carried out with HolonomicFunctions.
The article Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices by Christoph Koutschan studies random walks in certain lattices; these studies involved heavy computer calculations.
The Proof of George Andrews' and David Robbins' q-TSPP conjecture was a remarkable result by Christoph Koutschan, Manuel Kauers, and Doron Zeilberger, that settled a 25-years-old conjecture. The computations which established its computer proof were done by HolonomicFunctions.
The article "The integrals in Gradshteyn and Ryzhik. Part 18: Some automatic proofs" by Christoph Koutschan and Victor Moll uses HolonomicFunctions to deals with some integrals from the book by Gradshteyn and Ryzhik. The notebook GR18.nb (for Mathematica version 7) contains the computations for these examples.
HolonomicFunctions was used in The 1958 Pekeris-Accad-WEIZAC Ground-Breaking Collaboration that computed Ground States of Two-Electron Atoms (and its 2010 Redux) by Christoph Koutschan and Doron Zeilberger.
Ira Gessel's conjecture about the enumeration of certain random walks has been proven by the computer, using the package HolonomicFunctions. The corresponding article is Proof of Ira Gessel's Lattice Path Conjecture (Manuel Kauers, Christoph Koutschan, Doron Zeilberger), Proceedings of the National Academy of Sciences 106(28), pp. 11502-11505, July 2009.