Clausen's theorem and hypergeometric functions over finite fields

We prove a general identity for a hypergeometric function over a finite field Fq, where q is a power of an odd prime. A special case of this identity was proved by Greene and Stanton in 1986. As an application, we prove a finite field analogue of Clausen's theorem expressing a as the square of a . As another application, we evaluate an infinite family of over Fq at z=−1/8. This extends a result of Ono, who evaluated one of these in 1998, using elliptic curves.