Moments of Gaussian hypergeometric functions over finite fields

We prove explicit formulas for certain first and second moment sums of families of Gaussian hypergeometric functions n+1Fn, n > 1, over finite fields with q elements, where q is an odd prime. This enables us to find an estimate for the value 6F5(1). In addition, we evaluate certain second moments of traces of the family of Clausen elliptic curves in terms of the value 3F2(−1). These formulas also allow us to express the product of certain 2F1 and n+1Fn functions in terms of finite field Appell series which generalizes current formulas for products of 2F1 functions. We finally give closed form expressions for sums of Gaussian hypergeometric functions defined using different multiplicative characters