Representation of hypergeometric products of higher nesting depths in difference rings

TL;DR

This paper introduces a symbolic framework using difference rings to represent and simplify complex nested hypergeometric products, enabling zero-recognition and algebraic independence analysis for nested sums.

Contribution

It develops a novel difference ring-based method to rephrase nested hypergeometric products, facilitating their algebraic independence and zero-recognition.

Findings

Provides an algorithmic rephrasing of nested hypergeometric products.

Enables zero-recognition for nested hypergeometric expressions.

Integrates with existing symbolic summation algorithms for nested sums.

Abstract

A non-trivial symbolic machinery is presented that can rephrase algorithmically a finite set of nested hypergeometric products in appropriately designed difference rings. As a consequence, one obtains an alternative representation in terms of one single product defined over a root of unity and nested hypergeometric products which are algebraically independent among each other. In particular, one can solve the zero-recognition problem: the input expression of nested hypergeometric products evaluates to zero if and only if the output expression is the zero expression. Combined with available symbolic summation algorithms in the setting of difference rings, one obtains a general machinery that can represent (and simplify) nested sums defined over nested products.