A Non-Holonomic Systems Approach to Special Function Identities

TL;DR

This paper generalizes Zeilberger's creative telescoping method to non-holonomic special functions, enabling automated summation and integration for a broader class of mathematical objects like Stirling numbers, Bernoulli numbers, and polylogarithms.

Contribution

It introduces a novel approach that considers ideal dimensions in Ore algebras, extending the applicability of symbolic summation and integration beyond holonomic functions.

Findings

Unified framework for non-holonomic identities

Algorithms for summation and integration of new classes

Enhanced computer algebra capabilities

Abstract

We extend Zeilberger's approach to special function identities to cases that are not holonomic. The method of creative telescoping is thus applied to definite sums or integrals involving Stirling or Bernoulli numbers, incomplete Gamma function or polylogarithms, which are not covered by the holonomic framework. The basic idea is to take into account the dimension of appropriate ideals in Ore algebras. This unifies several earlier extensions and provides algorithms for summation and integration in classes that had not been accessible to computer algebra before.