Abstract
In this article, we first aim to give simple proofs of known formulae for the generalized Carlitz q-Bernoulli polynomials ÎČm,Ï(x, q) in the p-adic case by means of a method provided by Kim and then to derive a complex, analytic, two-variable q-L-function that is a q-analog of the two-variable L-function. Using this function, we calculate the values of two-variable q-L-functions at nonpositive integers and study their properties when q tends to 1. Mathematics Subject Classification (2000): 11B68; 11S80.
Highlights
When one talks of a q-extension, q can be variously considered as an indeterminate, a complex number q Ă C, or a p-adic number q Ă Cp
We shall provide some basic formulas for the generalized Carlitz q-Bernoulli polynomials which will be used to prove the main results of this article
Kim [12] defined a class of p-adic interpolation functions Gp,q(x) of the Carlitz q-Bernoulli numbers bm(q) and gave several interesting applications of these functions
Summary
We shall call them the Carlitz q-Bernoulli numbers and polynomials. Some properties of Carlitz q-Bernoulli numbers bm(q) were investigated by various authors. He constructed a complex analytic q-L-series that is a q-analog of Dirichlet L-function and interpolates Carlitz q-Bernoulli numbers, which is an answer to Koblitzâs question.
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