A quadratic hypergeometric 2F1 transformation over finite fields

TL;DR

This paper proves a conjectured quadratic transformation formula for hypergeometric 2F1 functions over finite fields, introducing a new linear transformation for pseudo hypergeometric functions and demonstrating its application.

Contribution

The paper establishes a previously conjectured quadratic transformation for hypergeometric functions over finite fields and introduces a novel linear transformation for pseudo hypergeometric functions.

Findings

Proof of the quadratic transformation conjecture

Development of a new linear transformation formula

Application demonstrating the utility of the transformation

Abstract

In 1984, the second author conjectured a quadratic transformation formula which relates two hypergeometric 2F1 functions over a finite field F_q. We prove this conjecture and give an application. The proof depends on a new linear transformation formula for pseudo hypergeometric functions over F_q.