Lauricella hypergeometric series $F_A^{(n)}$ over finite fields

TL;DR

This paper develops a finite field analogue of the Lauricella hypergeometric series $F_A^{(n)}$, extending classical results by expressing it through binomial coefficients, and deriving transformation, reduction formulas, and generating functions.

Contribution

It introduces a finite field analogue of the Lauricella series $F_A^{(n)}$, including new formulas and generating functions, expanding the theory over finite fields.

Findings

Finite field analogue of $F_A^{(n)}$ established

Transformation and reduction formulas derived

Generating functions for the series obtained

Abstract

In this paper, we develop a finite field analogue for one of the Lauricella series, $F^{(n)}_A$. Extending results of Greene, a finite field analog for the multinomial coefficient is developed in order to express the Lauricella series in terms of binomial coefficients. We have further deduced certain transformation and reduction formulas for the Lauricella series $F^{(n)}_A$. Finally, we have obtained a number of generating functions for the Lauricella series $F^{(n)}_A$.