q-supercongruences for truncated q-Appell series

Abstract

Congruences and q-congruences for the truncated Appell series are quite finite in the literature. In this paper, we introduce the truncated q-Appell series and investigate their congruence properties. Specifically, using two hypergeometric summations, the ‘creative microscoping’ method formulated by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish some q-supercongruences on the truncated q-Appell series ${\Phi }^{\left(1\right)},{\Phi }^{\left(2\right)},{\Phi }^{\left(3\right)}$. Moreover, in terms of the q-Zeilberger algorithm, a q-supercongruence for the truncated q-Appell series ${\Phi }^{\left(4\right)}$ is built, which is a new q-analogue of a conjecture of Apagodu and Zeilberger. As conclusions, we immediately obtain some congruences of the truncated Appell series.