Commentarii Mathematici Helvetici

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Volume 85, Issue 3, 2010, pp. 535–549
DOI: 10.4171/CMH/204

Published online: 2010-05-23

On the integro-differential equation satisfied by the p-adic log Γ-function

Eduardo Friedman[1]

(1) Universidad de Chile, Santiago de Chile, Chile

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

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

p y(x + t) dt = (x − 1) y′(x)
associated to (∗) is an infinite-dimensional commutative and associative p-adic algebra under the product law
(y1y2)(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.

Keywords: p-adic gamma function, p-adic integral equations, p-adic differential equations

Friedman Eduardo: On the integro-differential equation satisfied by the p-adic log Γ-function. Comment. Math. Helv. 85 (2010), 535-549. doi: 10.4171/CMH/204