# 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

^{[1]}(1) Universidad de Chile, Santiago de Chile, Chile

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

(∗) ∫

where ∫_{ℤp}`f`(`x`+`t`)`dt`= (`x`−1)`f`′(`x`) −`x`+ 1/2 (`x`∈ ℚ_{p}− ℤ_{p}),_{ℤp}is a Volkenborn integral and

`f`′ is the derivative of

`f`. We show that this equation characterizes log Γ

_{D}(

`x`) up to a function with everywhere vanishing second derivative. Namely, every solution

`f`of (∗) is infinitely differentiable and satisfies

`f`′′ = log Γ

_{D}′′.

We show that the set of solutions of the homogeneous equation

∫

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

(

the unit being `y`_{1}◊`y`_{2})(`x`) : =`y`_{2}′(`x`)`y`_{1}(`x`) +`y`_{1}′(`x`)`y`_{2}(`x`) − (`x`− 1/2)`y`_{1}′(`x`)`y`_{2}′(`x`),`y`(

`x`) =

`x`− 1/2. We also study Morita’s alternate

`p`-adic analogue log Γ

_{M}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