Evaluations of hypergeometric functions over finite fields (Q730692)

The authors continue their previous study and prove two general formulas for a two-parameter family of hypergeometric \({}_3F_2(z)\) functions over a finite field \(\mathbb F_q\), where \(q\) is a power of an odd prime. Each formula evaluates a \({}_3F_2\) in terms of a \({}_2F_1\) over \(\mathbb F_{q^2}\). As applications, they evaluate infinite one-parameter families of \({}_3F_2(\frac{1}{4})\) and \({}_3F_2(-1)\), thereby extending results of \textit{J. Greene} and \textit{D. Stanton} [J. Number Theory 23, 136--148 (1986; Zbl 0588.10038)] and \textit{K. Ono}, Trans. Am. Math. Soc. 350, No. 3, 1205--1223 (1998; Zbl 0910.11054)], who gave evaluations in special cases.