N T ] 3 0 N ov 2 00 6 Further Remarks on Multiple p-adic qL-Function of Two Variables ∗ (original) (raw)
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Further Remarks on Multiple p-adic q-L-Function of Two Variables ∗
2006
The object of this paper is to give several properties and applications of the multiple p-adic q-L-function of two variables L (r) p,q (s, z, χ). The explicit formulas relating higher order q- Bernoulli polynomials, which involve sums of products of higher order q-zeta function and higher order Dirichlet q-L-function are given. The value of higher order Dirichlet p-adic q-L-function for positive integers is also calculated. Furthermore, the Kummer-type congruences for multiple generalized q-Bernoulli polynomials are derived by making use of the difference theorem of higher order Dirichlet p-adic q-L-function.
Multiple two-variable p-adic q-L-function and its behavior at s = 0
Russian Journal of Mathematical Physics, 2008
The objective of this paper is to construct a multiple p-adic q-L-function of two variables which interpolates multiple generalized q-Bernoulli polynomials. By using this function, we solve a question of Kim and Cho. We also define a multiple partial q-zeta function which is related to the multiple q-L-function of two variables. Finally, we give a finite-sum representation of the multiple p-adic q-L-function of two variables and prove a multiple q-extension of the generalized formula of Diamond and Ferrero-Greenberg.
On the two-variable Dirichlet q-L-series
arXiv: Number Theory, 2005
In this study, we construct the two-variable multiple Dirichlet q-L-function and two-variable multiple Dirichlet type Changhee q-L-function. These functions interpolate the q-Bernoulli polynomials and generalized Changhee q-Bernoulli polynomials. By using the Mellin transformation, we give an integral representation for the two-variable multiple Dirichlet type q-zeta function and the two variable multiple Dirichlet type Changhee q-L-function.
Applied Mathematics and Computation, 2008
In this paper, we first investigate several further interesting properties of the multiple Hurwitz-Lerch Zeta function U n (z, s, a) which was introduced recently by Choi et al. [J. Choi, D.S. Jang, H.M. Srivastava, A generalization of the Hurwitz-Lerch Zeta function, Integral Transform. Spec. Funct., 19 (2008)]. We then introduce and investigate some q-extensions of the multiple Hurwitz-Lerch Zeta function U n (z, s, a), the Apostol-Bernoulli polynomials B ðnÞ k ðx; kÞ of order n, and the Apostol-Euler polynomials E ðnÞ k ðx; kÞ of order n. Relevant connections of the results presented here with those obtained in earlier works are also indicated precisely.
q-Hardy–Berndt type sums associated with q-Genocchi type zeta and q-l-functions
Nonlinear Analysis: Theory, Methods & Applications, 2009
The aim of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, q-analogue Genocchi type l-function and two-variable q-Genocchi type l-function. Furthermore, we construct new genereting functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations related to q-Hardy-Berndt type sums and q-Genocchi zeta function as well.
Applied Mathematics and Computation, 2009
a r t i c l e i n f o Keywords: q-Extensions of the Riemann zeta function and the Hurwitz zeta function q-Extensions of the Bernoulli and Euler polynomials and numbers q-Stirling numbers of the second kind Euler-Maclaurin summation formula a b s t r a c t In this paper, we systematically recover the identities for the q-eta numbers g k and the qeta polynomials g k ðxÞ, presented by Carlitz [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987-1000], which we define here via generating series rather than via the difference equations of Carlitz. Following a method developed by Kaneko et al. [M. Kaneko, N. Kurokawa, M. Wakayama, A variation of Euler's approach to the Riemann zeta function, Kyushu J. Math. 57 (2003) 175-192] for a canonical q-extension of the Riemann zeta function, we investigate a similarly constructed q-extension of the Hurwitz zeta function. The details of this investigation disclose some interesting connections among q-eta polynomials, Carlitz's q-Bernoulli polynomials b k ðxÞ;-polynomials, and the q-Bernoulli polynomials that emerge from the q-extension of the Hurwitz zeta function discussed here.
TURKISH JOURNAL OF MATHEMATICS
In this paper, by applying the p-adic q-integrals to a family of continuous differentiable functions on the ring of p-adic integers, we construct new generating functions for generalized Apostol-type numbers and polynomials attached to the Dirichlet character of a finite abelian group. By using these generating functions with their functional equations, we derive various new identities and relations for these numbers and polynomials. These results are generalizations of known identities and relations including some well-known families of special numbers and polynomials such as the generalized Apostol-type Bernoulli, the Apostol-type Euler, the Frobenius-Euler numbers and polynomials, the Stirling numbers, and other families of numbers and polynomials. Moreover, by the help of these generating functions, we also construct other new families of numbers and polynomials with their generating functions. By using these functions, we investigate some fundamental properties of these numbers and polynomials. Finally, we also give explicit formulas for computing the Apostol-Bernoulli and Apostol-Euler numbers.
New families of special numbers and polynomials arising from applications of p-adic q-integrals
Advances in Difference Equations
In this manuscript, generating functions are constructed for the new special families of polynomials and numbers using the p-adic q-integral technique. Partial derivative equations, functional equations and other properties of these generating functions are given. With the help of these equations, many interesting and useful identities, relations, and formulas are derived. We also give p-adic q-integral representations of these numbers and polynomials. The results we have obtained for these special numbers and polynomials are closely related to well-known families of polynomials and numbers including the Bernoulli numbers, the Apostol-type Bernoulli numbers and polynomials and the Frobenius-Euler numbers, the Stirling numbers, and the Daehee numbers. We give some remarks and observations on the results of this paper.
Cogent Mathematics
By applying the p-adic q-Volkenborn Integrals including the bosonic and the fermionic p-adic integrals on p-adic integers, we define generating functions, attached to the Dirichlet character, for the generalized Apostol-Bernoulli numbers and polynomials, the generalized Apostol-Euler numbers and polynomials, generalized Apostol-Daehee numbers and polynomials, and also generalized Apostol-Changhee numbers and polynomials. We investigate some properties of these numbers and polynomials with their generating functions. By using these generating functions and their functional equation, we give some identities and relations including the generalized Apostol-Daehee and Apostol-Changhee numbers and polynomials, the Stirling numbers, the Bernoulli numbers of the second kind, Frobenious-Euler polynomials, the generalized Bernoulli numbers and the generalized Euler numbers and the Frobenious-Euler polynomials. By using the bosonic and the fermionic p-adic integrals, we derive integral represantations for the generalized Apostol-type Daehee numbers and the generalized Apostol-type Changhee numbers.