Certain transformations and special values of hypergeometric functions over finite fields
\textit{J. Greene} [Trans. Am. Math. Soc. 301, 77--101 (1987; Zbl 0629.12017)] introduced hypergeometric functions over finite fields, which are also known as Gaussian hypergeometric series, and in particular deduced certain special values of \(_2F_1\) and \(_3F_2\) hypergeometric functions. Since then many special values of hypergeometric functions over finite fields are obtained using different techniques. Most of the known results give values of \(_2F_1\) and \(_3F_2\) hypergeometric functions containing trivial, quadratic and quartic characters as parameters. Finding values of Gaussian hypergeometric series containing arbitrary characters at specific values of the argument is a difficult problem and only a few such values are known till now. It is remarkable that special values of Gaussian hypergeometric series play an important role in solving many old conjectures and supercongruences. \textit{K. Ono} [Trans. Am. Math. Soc. 350, No. 3, 1205--1223 (1998; Zbl 0910.11054)] explicitly evaluated special values of \(_2F_1\) and \(_3F_2\) hypergeometric functions containing trivial and quadratic characters by counting points on elliptic curves with complex multiplication over \({\mathbb F}_p\). By relating the values of \(_3F_2\) hypergeometric functions obtained by Ono to squares of certain \(_2F_1\) hypergeometric functions, Kalita evaluated special values of squares of certain \(_2F_1\) hypergeometric functions containing trivial and quadratic characters. In the current work, Barman and Tripathi find finite field analogues of certain transformations satisfied by the classical hypergeometric series. Using properties of Gauss and Jacobi sums, they evaluate certain character sums to establish these transformations. They then use their new transformation formulas to find several special values of Gaussian hypergeometric series including special values of \(_2F_1\) hypergeometric functions containing arbitrary characters at certain arguments. Certain special values of \(_2F_1\) hypergeometric functions over finite fields containing trivial and quadratic characters obtained by Ono follow from the authors' special values of \(_2F_1\) hypergeometric functions containing arbitrary characters as parameters. Also one of the finite field analogues of algebraic hypergeometric identities given by Fuselier, Long, Ramakrishna, Swisher, and Tu follows from one of the authors' transformation formulas in the current article.
hypergeometric functions over finite fields
scientific article; zbMATH DE number 7493917
scientific article