$q$-Bernoulli Numbers and Polynomials Associated with Multiple $q$-Zeta Functions and Basic $L$-series

By Taekyun Kim, Yilmaz Simsek, H. Srivastav. 2005

207 citations recorded by the source corpus.

By using $q$-Volkenborn integration and uniform differentiable on $\mathbb{Z}%_{p}$, we construct $p$-adic $q$-zeta functions. These functions interpolate the $q$-Bernoulli numbers and polynomials. The value of $p$-adic $q$-zeta functions at negative integers are given explicitly. We also define new generating functions of $q$-Bernoulli numbers and polynomials. By using these functions, we prove analytic continuation of some basic (or $q$-) $L$% -series. These generating functions also interpolate Barnes' type Changhee $% q $-Bernoulli numbers with attached to Dirichlet character as well. By applying Mellin transformation, we obtain relations between Barnes' type $q$% -zeta function and new Barnes' type Changhee $q$-Bernolli numbers. Furthermore, we construct the Dirichlet type Changhee (or $q$-) $L$% -functions.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/10ee8e84ea6667a703bebc87bddb5e44eaf90755

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In Algebra and Number Theory

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Citations in the corpus

A Note on the-Genocchi Numbers and Polynomials (2007, methodology): “0≤x 12 ] ). In the previous paper [4] author constructed the q-extension of Euler polynomials in the fermionic sense of p-adic q-integral at q = −1 as follows:”
The Modified q-Euler numbers and polynomials (2007, background): “1 + q = 1 − q + q 2 − � � � + (−q)n−1, cf. [4,5,7, 9 ].”
New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials (2007, methodology): “If h = 1, limw→1 B (v) n (q) = B (v) n (q) (see [27]).”
q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients (2008, background): “Carlitz’s q-Bernoulli numbers βk,q can be defined recursively by β0,q = 1 and by the rule that q(qβ +1 ) k − βk,q is equal to 1i fk = 1 and to 0i f k> 1 with the usual convention of replacing β i by βi,q (see [2, 3, 24, 25 ]).”
q-Generalized Euler numbers and polynomials (2006, background): “Carlitz’s q-Bernoulli numbers βk can be determined inductively by β0 = 1, q(qβ + 1) k − βk = { 1 if k = 1 0 if k > 1, (1) with the usual convention about replacing β by βi,q(see [1,2,3,4,5,12]).”
Explicit formulas for p-adic integrals: approach to p-adic distributions and some families of special numbers and polynomials. (2019, background): “439], [124]; see also the references cited in each of these earlier works).”
Note on the Euler Numbers and Polynomials (2008, background): “with the usual convention about replacing � i by �i,q(see [1,2,3,4,5, 12 ]).”
Complete sum of products of (h, q)-extension of Euler polynomials and numbers (2010, background): “Bernoulli and Euler numbers and polynomials were studied by many authors (see for detail [18], [19], [9], [7], [10], [11], [12], [13], [16], [8], [20], [21], [22], [ 24 ], [14]) . We introduce some of them here.”