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

In [T. Kim, S.H. Rim, Generalized Carlitz’s qq-Bernoulli numbers in the pp-adic number field, Adv. Stud. Contemp. Math. 2 (2000) 9–19], the new qq-extension of Bernoulli polynomials and generalized Bernoulli numbers attached to χχ were constructed by using pp-adic invariant integral on ZpZp. In this paper we construct the new qq-extension of generalized Bernoulli polynomials attached to χχ due to author and derive the existence of a specific pp-adic interpolation function which interpolates the qq-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.