Congruences corresponding to hypergeometric identities II. Generalized hypergeometric transformations

Abstract

We consider the congruences modulo $p^2$ for the truncated generalized hypergeometric functions ${}_{r}F_s{\left[\begin{array}{c}{\alpha }_1\\ \end{array}\begin{array}{c}{\alpha }_2\\ {\beta }_1\end{array}\begin{array}{c}\dots \\ \dots \end{array}\begin{array}{c}{\alpha }_r\\ {\beta }_s\end{array}\right|1]}_n$. In particular, we give the $\mathrm{mod}{p}^2$ analogue of Whipple's ${}_{4}F_3$ and ${}_{7}F_6$ transformations.