On the differential transcendentality of the Morita p-adic gamma function

Abstract

In p-adic analysis one can find an analog of the classical gamma function defined on the ring of p-adic integers. In 1975, Morita defined the p-adic gamma function Gamma _p by a suitable modification of the function n mapsto n!. In this note we prove that for any given prime number p the Morita p-adic gamma function Gamma _p is differentially transcendental over {mathbb {C}}_p(X). The main result is an analog of the classical Hölder’s theorem, which states that Euler’s gamma function Gamma does not satisfy any algebraic differential equation whose coefficients are rational functions.