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.
Categories: math.NT