Transformations of the hypergeometric 4F3 with one unit shift: a group theoretic study

TL;DR

This paper investigates the group structure of hypergeometric 4F3 transformations with a unit shift, revealing their algebraic properties, subgroup structures, and relations to 3F2 transformations, supported by computational tools.

Contribution

It characterizes the transformation group of 4F3 functions with a unit shift, identifies a specific subgroup structure, and connects these to 3F2 transformations, providing new computational methods.

Findings

The transformation group is explicitly characterized and shown to have a specific algebraic structure.

A subgroup isomorphic to the direct product of S_5 and Z^5 is identified and studied.

New summation formulas and computational routines for group calculations are provided.

Abstract

We study the group of transformations of 4F3 hypergeometric functions evaluated at unity with one unit shift in parameters. We reveal the general form of this family of transformations and its group property. Next, we use explicitly known transformations to generate a subgroup whose structure is then thoroughly studied. Using some known results for 3F2 transformation groups, we show that this subgroup is isomorphic to the direct product of the symmetric group of degree 5 and 5-dimensional integer lattice. We investigate the relation between two-term 4F3 transformations from our group and three-term 3F2 transformations and present a method for computing the coefficients of the contiguous relations for 3F2 functions evaluated at unity. We further furnish a class of summation formulas associated with the elements of our group. In the appendix to this paper, we give a collection of Wolfram Mathematica routines facilitating the group calculations.