HolonomicFunctions: A Mathematica package for dealing with multivariate holonomic functions, including closure properties, summation, and integration

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.


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.

The HolonomicFunctions package is registered in swMATH, where a more extensive list of papers using and citing the package can be found.