Extensions of Karlsson-Minton summation theorem and some consequences of the first Miller-Paris transformation

TL;DR

This paper extends the Karlsson-Minton summation theorem for hypergeometric functions, explores the Miller-Paris transformation, and introduces new summation and transformation formulas with broader parameter ranges.

Contribution

It provides new independent extensions of the Karlsson-Minton formula and explores alternative forms of the Miller-Paris transformation, broadening their applicability.

Findings

Extended the Karlsson-Minton summation formula to include prohibited parameter values.

Established a recurrence relation linking integer negative differences to the unit negative difference case.

Presented alternative forms of the Miller-Paris transformation, including one with Meijer-Norlund G functions.

Abstract

In this paper we give several independent extensions of the Karlsson-Minton summation formula for the generalized hypergeometric function with integral parameters differences. In particular, we examine the "prohibited" values for the integer top parameter in Minton's formula, extend one unit negative difference in Karlsson's formula to a finite number of integer negative differences and establish known and new summation and transformation formulas when the unit negative difference is allowed to take arbitrary values. We also present a recurrence relation reducing the case of integer negative difference to the Karlsson-Minton case of unit negative difference. Further, we explore some alternative forms of the first Miller-Paris transformation, including one expressed in terms of Meijer-Norlund G function.