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

We study a special case of Kashio’s $p$-adic $\log \mathrm{\Gamma }$-function, that we call $\mathrm{L}\mathrm{o}\mathrm{g}{\mathrm{\Gamma }}_p$, 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 $\mathrm{L}\mathrm{o}\mathrm{g}{\mathrm{\Gamma }}_p$ which generalizes and links the known distribution formulas for Diamond’s and Morita’s functions.