A note on Carlitz q-Bernoulli numbers and polynomials

Abstract

In this article, we first aim to give simple proofs of known formulae for the generalized Carlitz q-Bernoulli polynomials b m,c (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-Lfunction 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.

New results on the q-generalized Bernoulli polynomials of level m

“…More recently, J. Choi, T. Ernst, D. kim, S. Nalci, C.S. Ryoo [3][4][5][6][7][8] defined the q-Bernoulli polynomials using different methods and studied their properties. There are numerous recent investigations on qgeneralizations of this subject by many others author; see [9][10][11][12][13][14][15][16][17].…”

Modified Degenerate Carlitz&rsquo;s <em>q</em>-Bernoulli Polynomials and Numbers with Weight (<em>&alpha;</em>, <em>&beta;</em>)

“…from which one can obtain the Bernoulli numbers B n as values B n (0) := B n (see, [2][3][4][5][6][7][8][9][10][11][12][13][14][15]). Carlitz [2] defined the q-analogue of the Bernoulli numbers β n = β n (q) and polynomials β n (x : q) as follows In recent years, q-Bernoulli polynomials and their generalizations have been studied and investigated extensively by many mathematicians.…”

“…A more detailed statement of the above is found in each of the references [1,[4][5][6][7][8][9][10][11][12][13][14]. The Carlitz's q-Bernoulli polynomials with Witt's formula are defined by the following p-adic q-integral on Z p , with respect to µ q (see [5]…”

Some results for Carlitz’s q-Bernoulli numbers and polynomials

“…Since the above Carlitz's q-Bernoulli numbers and q-Bernoulli polynomials first appeared, different properties for them have been well studied by many authors; see, for example, [18,19,20,31,33]. In fact, Carlitz's q-Bernoulli numbers and polynomials can be defined by the following exponential generating functions (see, e.g., [24,27]): If the left-hand side of (1.6) is denoted by F q (t, x) then the Mellin transform gives…”