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

Eduardo Friedman

doi:10.4171/cmh/204

JournalscmhVol. 85, No. 3pp. 535–549

On the integro-differential equation satisfied by the $p$ -adic $lo g Γ$ -function

Eduardo Friedman
Universidad de Chile, Santiago de Chile, Chile

$On the integro-differential equation satisfied by the $p$-adic $\log\, Γ$-function cover$

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Abstract

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

\int_{Z_{p}} f (x + t) d t = (x - 1) f' (x) - x + 1/2 (x \in Q_{p} - Z_{p}), (*)

where $\int_{Z_{p}}$ is a Volkenborn integral and $f^{'}$ is the derivative of $f$ . We show that this equation characterizes $lo g Γ D (x)$ up to a function with everywhere vanishing second derivative. Namely, every solution $f$ of (∗) is infinitely differentiable and satisfies $f^{''} = lo g Γ D^{''}$ .

We show that the set of solutions of the homogeneous equation

\int_{Z_{p}} y (x + t) d t = (x - 1) y^{'} (x)

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

(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),

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

Cite this article

Eduardo Friedman, On the integro-differential equation satisfied by the $p$ -adic $lo g Γ$ -function. Comment. Math. Helv. 85 (2010), no. 3, pp. 535–549

DOI 10.4171/CMH/204