# On the integro-differential equation satisfied by the $p$-adic $gΓ$-function

### Eduardo Friedman

Universidad de Chile, Santiago de Chile, Chile

## Abstract

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

where $∫_{Z_{p}}$ is a Volkenborn integral and $f_{′}$ is the derivative of $f$. We show that this equation characterizes $gΓD(x)$ up to a function with everywhere vanishing second derivative. Namely, every solution $f$ of (∗) is infinitely differentiable and satisfies $f_{′′}=gΓD_{′′}$.

We show that the set of solutions of the homogeneous equation

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

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

## Cite this article

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

DOI 10.4171/CMH/204