# A distribution formula for Kashio's $p$-adic log-gamma function

## TL;DR

This paper introduces a new distribution formula for Kashio's $p$-adic log-gamma function, unifying and extending existing formulas by Morita and Diamond, and emphasizing its local analyticity.

## Contribution

It provides a novel distribution formula for Kashio's $p$-adic log-gamma function that generalizes previous results and highlights its local analytic properties.

## Key findings

- Unified distribution formula for Kashio's $p$-adic log-gamma function
- Generalizes known formulas for Diamond's and Morita's functions
- Establishes local analyticity of the new function

## Abstract

We study a special case of Kashio's $p$-adic $\log\Gamma$-function, that we call ${\mathrm{Log}}\Gamma_{\mathit{p}}$, which combines these of Morita and Diamond. It agrees with each of these on large parts of its domain and has the advantage of being a locally analytic function. We prove a distribution formula for ${\mathrm{Log}}\Gamma_{\mathit{p}}$ which generalizes and links the known distribution formulas for Diamond's and Morita's functions.