Alternative approach to Miller-Paris transformations and their extensions

TL;DR

This paper introduces a new derivation of Miller-Paris transformations for hypergeometric functions using generalized Stieltjes transforms, resulting in simpler forms and extensions with additional parameters.

Contribution

It provides an independent derivation of Miller-Paris transformations employing Stieltjes transforms, simplifying characteristic polynomials and extending the transformations to more general cases.

Findings

New, simpler forms of characteristic polynomials for Miller-Paris transformations

Extended transformations to include additional free parameters

Derived degenerate versions for cases with negative integer parameter differences

Abstract

Miller-Paris transformations are extensions of Euler's transformations for the Gauss hypergeometric functions to generalized hypergeometric functions of higher-order having integral parameter differences (IPD). In our recent work we computed the degenerate versions of these transformations corresponding to the case when one parameter difference is equal to a negative integer. The purpose of this paper is to present an independent new derivation of both the general and the degenerate forms of Miller-Paris transformations. In doing so we employ the generalized Stieltjes transform representation of the generalized hypergeometric functions and some partial fraction expansions. Our approach leads to different forms of the characteristic polynomials, one of them appears noticeably simpler than the original form due to Miller and Paris. We further present two extensions of the degenerate transformations to the generalized hypergeometric functions with additional free parameters and additional parameters with negative integral differences.