On the q-analogue of two-variable p-adic L-function

We construct the two-variable $p$-adic $q$-$L$-function which interpolates the generalized $q$-Bernoulli polynomials associated with primitive Dirichlet character $\chi$. Indeed, this function is the $q$-extension of two-variable $p$-adic $L$-function due to Fox, corresponding to the case $q=1 .$ Finally, we give some $p$-adic integral representation for this two-variable $p$-adic $q$-$L$-function and derive to $q$-extension of the generalized formula of Diamond and Ferro and Greenberg for the two-variable $p$-adic $L$-function in terms of the $p$-adic gamma and $\log$ gamma function.