Certain transformations and special values of hypergeometric functions over finite fields

Abstract

In this article we find finite field analogues of certain transformations satisfied by the classical hypergeometric series. Using properties of Gauss and Jacobi sums we evaluate certain character sums to establish these transformations. We then use these transformations to evaluate explicitly several special values of $${_2}F_1$$ hypergeometric functions over finite fields. Certain special values of $${_2}F_1$$ hypergeometric functions over finite fields containing trivial and quadratic characters obtained by Ono follow from our special values of $${_2}F_1$$ hypergeometric functions containing arbitrary characters as parameters. One of the finite field analogues of algebraic hypergeometric identities given by Fuselier, Long, Ramakrishna, Swisher, and Tu also follows from one of our transformation formulas.