Appell series over finite fields and Gaussian hypergeometric series

Abstract

In this article, we find finite field analogues of certain identities satisfied by the classical $${_4}F_3$$ -hypergeometric series and Appell series. As an application, we find new summation and product formulas satisfied by the Gaussian hypergeometric series. For example, we express a $${_4}F_3$$ -Gaussian hypergeometric series as a sum of two $${_2}F_1$$ -Gaussian hypergeometric series. We also find two identities expressing $${_4}F_3$$ -Gaussian hypergeometric series as a product of two $${_2}F_1$$ -Gaussian hypergeometric series. As another application, we find a special value of a $${_4}F_3$$ -Gaussian hypergeometric series using the summation formula.