Supercongruences Arising from Truncated Appell Series

Abstract

In recent years, both Appell series and truncated Appell series have garnered significant research interest among scholars. In this paper, we investigate the congruence properties of truncated Appell series and truncated $q$-Appell series. By employing various combinatorial identities and the `creative microscoping' method introduced by Guo and Zudilin, we provide $q$-analogues for several known congruences of truncated Appell series $F_1$ and $F_2$. Meanwhile, we establish a new $q$-supercongruence related to the truncated Appell series $F_4$ and propose a related conjecture, which provides a $q$-analogue for two conjectures by Apagodu and Zeilberger.