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

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.