Degenerate Miller-Paris transformations

TL;DR

This paper extends the theory of Miller-Paris transformations for hypergeometric functions by deriving new formulas for cases previously considered invalid, thus filling a significant gap in the existing mathematical framework.

Contribution

It computes the limit cases of Miller-Paris transformations when the free bottom parameter exceeds the free top parameter by a small positive integer, introducing new transformation and summation formulas.

Findings

Derived new transformation formulas for previously excluded parameter cases

Extended the Karlsson-Minton theorem with new summation formulas

Provided a comprehensive analysis of limit cases in hypergeometric transformations

Abstract

Important new transformations for the generalized hypergeometric functions with integral parameter differences have been discovered some years ago by Miller and Paris and studied in detail in a series of papers by a number of authors. These transformations fail if the free bottom parameter is greater than a free top parameter by a small positive integer. In this paper we fill this gap in the theory of Miller-Paris transformations by computing the limit cases of these transformations in such previously prohibited situations. This leads to a number of new transformation and summation formulas including extensions of Karlsson-Minton theorem.