$p$-adic hypergeometric function related with $p$-adic multiple polylogarithms

Hidekazu Furusho

Published 2022-11-14Version 1

This paper introduces a $p$-adic analogue of Gauss's hypergeometric function which is constructed along ideas different from Dwork's. Our construction is based on a relationship between the hypergeometric function and the KZ equation. We perform a residue-wise analytic prolongation of our $p$-adic hypergeometric function by investigating its relationship with $p$-adic multiple polylogarithms. By further exploring its local behavior around the point $1$, we show that a $p$-adic analogue of Gauss hypergeometric theorem holds for the function.