Evaluations of hypergeometric functions over finite fields

We prove two general formulas for a two-parameter family of hypergeometric \3F2(z) functions over a finite field \F_q, where q is a power of an odd prime. Each formula evaluates a \3F2 in terms of a \2F1 over \F_{q^2}. As applications, we evaluate infinite one-parameter families of \3F2(\frac{1}{4}) and \3F2(-1), thereby extending results of J. Greene--D. Stanton and K. Ono, who gave evaluations in special cases.