## On $$p$$-adic twisted $$q$$-$$L$$-functions related to generalized twisted Bernoulli numbers.(English)Zbl 1163.11312

Summary: In this paper, we construct a twisted $$q$$-partial zeta function and some twisted two-variable $$q$$-$$L$$-functions that interpolate $$q$$-Bernoulli numbers, $$\beta _{ n,\xi } ^{(h)} (q)$$, and Bernoulli polynomials, $$\beta _{ n,x,\xi } ^{(h)} (x, q)$$, respectively, at negative integers. Using these functions, we prove the existence of a $$p$$-adic interpolation function that interpolates the $$q$$-generalized polynomials $$\beta _{ n,x,\xi } ^{ (h) } (x, q)$$ at negative integers. Consequently, we define a $$p$$-adic twisted $$q$$-$$L$$-function which is a solution of a question of T. Kim et al. [Far East J. Appl. Math. 13, No. 1, 13–21 (2003; Zbl 1046.11009)].

### MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 11B68 Bernoulli and Euler numbers and polynomials 11S40 Zeta functions and $$L$$-functions 33D99 Basic hypergeometric functions

Zbl 1046.11009
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