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This package is part of the RISCErgoSum bundle. See Download and Installation.
Short Description¶
The HolonomicFunctions package 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.
Accompanying Files¶
Right now you are using Version 1.7.3 released on March 17, 2017. This
version is compatible with Mathematica versions from 5.2 to 11.0.
Please report any bugs and comments to Christoph Koutschan.
Literature¶
The theoretical background of the algorithms implemented in
HolonomicFunctions and how to use the package, is described in
C. Koutschan,
Advanced Applications of the Holonomic Systems Approach,
RISC, Johannes Kepler University, Linz. PhD Thesis. September 2009.
[pdf] C. Koutschan,
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 in
detail described in the documentation
C. Koutschan,
HolonomicFunctions (User’s Guide),
Technical report no. 10-01 in RISC Report Series,
University of Linz, Austria. January 2010.
[pdf]
Some Applications¶
The package HolonomicFunctions has been applied in many different
contexts, some of which are listed below.
In Inverse inequality estimates with symbolic computation
the HolonomicFunctions package was used to evaluate holonomic
determinants that arose in numerical analysis. By means of the holonomic gradient method, the HolonomicFunctions
package contributed to the analysis of wireless communication
networks, as described in the papers MIMO zero-forcing performance
evaluation using the holonomic gradient method
and Exact ZF analysis and computer-algebra-aided evaluation in rank-1
LoS Rician fading. 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. In physics, the HolonomicFunctions package has contributed to the
evaluation of relativistic Coulomb integrals
and to the study of fundamental laser modes in paraxial optics. In numerical analysis, finite element methods are used to construct
approximate solutions to partial differential equations. In some
instances, HolonomicFunctions was able to derive the
differential-difference relations between the basis functions,
that are necessary for an efficient implementation for Maxwell’s equations;
this work finally led to a registered patent. 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. This work has been awarded the David P. Robbins
Prize 2016 of the American Mathematical Society. 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.
The HolonomicFunctions package is registered in
swMATH,
where a more extensive list of papers using and citing the
package can be found.
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