Exponential sums and finite field $A$-hypergeometric functions

TL;DR

This paper introduces finite field $A$-hypergeometric functions, demonstrating their role as Fourier expansions of exponential sums on the torus, and connecting them to McCarthy's finite field hypergeometric functions.

Contribution

It defines a new class of finite field hypergeometric functions and links them to existing hypergeometric functions through specialization.

Findings

Finite field $A$-hypergeometric functions are Fourier expansions of exponential sums.

They can be specialized to McCarthy's finite field hypergeometric functions.

The framework unifies different hypergeometric functions over finite fields.

Abstract

We define finite field $A$-hypergeometric functions and show that they are Fourier expansions of families of exponential sums on the torus. For an appropriate choice of $A$, our finite field $A$-hypergeometric function can be specialized to the finite field ${}_kF_{k-1}$-hypergeometric function defined by McCarthy.