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This paper introduces a $p$-adic analogue of Gauss's hypergeometric functionwhich is constructed along ideas different from Dwork's. Our construction isbased on a relationship between the hypergeometric function and the KZequation. We perform a residue-wise analytic prolongation of our $p$-adichypergeometric function by investigating its relationship with $p$-adicmultiple polylogarithms. By further exploring its local behavior around thepoint $1$, we show that a $p$-adic analogue of Gauss hypergeometric theoremholds for the function.

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