Duality relations for hypergeometric series

TL;DR

This paper establishes algebraic duality relations for hypergeometric and q-hypergeometric solutions, providing a broad family of identities with roots in classical analysis but derived through modern algebraic methods.

Contribution

It introduces a purely algebraic framework for duality relations in hypergeometric and q-hypergeometric equations, extending classical analytic insights.

Findings

Derived explicit duality relations for hypergeometric solutions.

Extended duality identities to q-hypergeometric equations.

Provided an algebraic approach based on differential and difference modules.

Abstract

We explicitly give the relations between the hypergeometric solutions of the general hypergeometric equation and their duals, as well as similar relations for q-hypergeometric equations. They form a family of very general identities for hypergeometric series. Although they were foreseen already by N. M. Bailey in the 1930's on analytic grounds, we give a purely algebraic treatment based on general principles in general differential and difference modules.