Holonomic Tools for Basic Hypergeometric Functions

TL;DR

This paper advocates for using holonomic systems and computer algebra tools to analyze basic hypergeometric functions, demonstrating their effectiveness through case studies and a new proof of a key q-analog formula.

Contribution

It introduces the application of holonomic tools to q-series and hypergeometric functions, including a novel computer-assisted proof of the Ismail-Zhang formula.

Findings

Successful application of holonomic tools to q-series

New computer-assisted proof of the Ismail-Zhang formula

Enhanced understanding of hypergeometric functions using software

Abstract

With the exception of q-hypergeometric summation, the use of computer algebra packages implementing Zeilberger's "holonomic systems approach" in a broader mathematical sense is less common in the field of q-series and basic hypergeometric functions. A major objective of this article is to popularize the usage of such tools also in these domains. Concrete case studies showing software in action introduce to the basic techniques. An application highlight is a new computer-assisted proof of the celebrated Ismail-Zhang formula, an important q-analog of a classical expansion formula of plane waves in terms of Gegenbauer polynomials.