Congruence relations for $p$-adic hypergeometric functions $\widehat{\mathscr{F}}_{a,...,a}^{(σ)}(t)$ and its transformation formula

Abstract

We introduce new kind of $p$-adic hypergeometric functions. We show these functions satisfy congruence relations, so they are convergent functions. And we show that there is a transformation formula between our new $p$-adic hypergeometric functions and $p$-adic hypergeometric functions of logarithmic type defined by M. Asakura in a particular case.