Exponential sums and finite field $A$-hypergeometric functions

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 ${}_k{F}_{k-1}$-hypergeometric function defined by McCarthy.