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

Abstract

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