On a -adic interpolation function for the -extension of the generalized Bernoulli polynomials and its derivative

In [T. Kim, S.H. Rim, Generalized Carlitz’s q -Bernoulli numbers in the p -adic number field, Adv. Stud. Contemp. Math. 2 (2000) 9–19], the new q -extension of Bernoulli polynomials and generalized Bernoulli numbers attached to χ were constructed by using p -adic invariant integral on Z p . In this paper we construct the new q -extension of generalized Bernoulli polynomials attached to χ due to author and derive the existence of a specific p -adic interpolation function which interpolates the q -extension of generalized Bernoulli polynomials at negative integers. Finally, we give the values of partial derivative for this function and investigate some properties which are related to this interpolation function.