On the integro-differential equation satisfied by the <var>p</var>-adic log Γ-function

Abstract

Diamond’s p-adic analogue log ΓD(x) of the classical function log Γ(x) has recently been shown to satisfy the integro-differential equation

(∗)   ∫ℤp f(x + t) dt = (x −1) f′(x) − x + 1/2   (x ∈ ℚp − ℤp),

where ∫ℤp is a Volkenborn integral and f′ is the derivative of f. We show that this equation characterizes log ΓD(x) up to a function with everywhere vanishing second derivative. Namely, every solution f of (∗) is infinitely differentiable and satisfies f′′ = log ΓD′′.

We show that the set of solutions of the homogeneous equation

∫ℤp y(x + t) dt = (x − 1) y′(x)

associated to (∗) is an infinite-dimensional commutative and associative p-adic algebra under the product law

(y1 ◊ y2)(x) : = y2′(x)y1(x) + y1′(x)y2(x) − (x − 1/2) y1′(x)y2′(x),

the unit being y(x) = x − 1/2. We also study Morita’s alternate p-adic analogue log ΓM of log Γ(x) and prove similar results.