Transformations of Well-Poised Hypergeometric Functions over Finite Fields

TL;DR

This paper introduces a hypergeometric function over finite fields, establishing transformation and summation formulas that mirror classical hypergeometric identities, and explores its connections to existing finite field analogues.

Contribution

It defines a new finite field hypergeometric function and proves several transformation and summation formulas analogous to classical identities.

Findings

Proved multiple transformation formulas for the finite field hypergeometric function.

Established summation formulas similar to classical hypergeometric series.

Explored relationships with Greene and Katz finite field hypergeometric functions.

Abstract

We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this function's relationship to other finite field analogues of the classical series, most notably those defined by Greene and Katz.