q-Bernoulli Numbers Research Articles

Some Identities with Multi-Generalized q-Hyperharmonic Numbers of Order r

The main purpose of this paper is to define multiple alternative q-harmonic numbers, Hnk;q and multi-generalized q-hyperharmonic numbers of order r, Hnrk;q by using q-multiple zeta star values (q-MZSVs). We obtain some finite sum identities and give some applications of them for certain combinations of q-multiple polylogarithms Liq;k1,k2,…,kdt1,t2,…,td with the help of generating functions. Additionally, one of the applications is the sum involving q-Stirling numbers and q-Bernoulli numbers.

On generalized q-poly-Bernoulli numbers and polynomials

Many mathematicians in ([1],[2],[5],[14],[20]) introduced and investigated the generalized q-Bernoulli numbers and polynomials and the generalized q-Euler numbers and polynomials and the generalized q-Gennochi numbers and polynomials. Mahmudov ([15],[16]) considered and investigated the q-Bernoulli polynomials B(?)n,q(x,y) in x,y of order ? and the q-Euler polynomials E(?) n,q (x,y)in x,y of order ?. In this work, we define generalized q-poly-Bernoulli polynomials B[k,?] n,q (x,y) in x,y of order ?. We give new relations between the generalized q-poly-Bernoulli polynomials B[k,?] n,q (x,y) in x,y of order ? and the generalized q-poly-Euler polynomials ?[k,?] n,q (x,y) in x,y of order ? and the q-Stirling numbers of the second kind S2,q(n,k).

Note on Type 2 Degenerate q-Bernoulli Polynomials

The purpose of this paper is to introduce and study type 2 degenerate q-Bernoulli polynomials and numbers by virtue of the bosonic p-adic q-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, identities regarding Carlitz’s q-Bernoulli numbers, identities concerning degenerate q-Bernoulli numbers, and the representations of the fully degenerate type 2 Bernoulli numbers in terms of moments of certain random variables, created from random variables with Laplace distributions. It is expected that, as was done in the case of type 2 degenerate Bernoulli polynomials and numbers, we will be able to find some identities of symmetry for those polynomials and numbers.

On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found.

On modified degenerate Carlitz q-Bernoulli numbers and polynomials

In a recent study by Kim (Bull. Korean Math. Soc. 53(4):1149-1156, 2016) an attempt was made to examine some of the identities and properties that are related to the degenerate Carlitz q-Bernoulli numbers and polynomials. In our paper we define the modified degenerate q-Bernoulli numbers and polynomials. As part of this we investigate some of the identities and properties that are associated with these numbers and polynomials which are derived from the generating functions and p-adic integral equations.

Symmetric identities for Carlitz's generalized twisted q-Bernoulli numbers and polynomials associated with p-adic q-integral on Zp

In this paper, we study the symmetry for the generalized twisted qBernoulli numbers n;;q; and polynomials n;;q; (x). We obtain some interesting identities of the power sums and the generalized twisted qBernoulli polynomials n;;q; (x) using the symmetric properties for the p-adic invariant q-integral on Zp. Mathematics Subject Classication: 11B68, 11S40, 11S80

On the Modified q-Euler Numbers and Polynomials withWeak Weight 0

In this paper, we construct new q-extension of Euler polynomials with weight 0. These modified q-Euler polynomials are useful to study various identities of Carlitz's q-Bernoulli numbers.

Some identities of q-Euler polynomials arising from q-umbral calculus

Recently, Araci-Acikgoz-Sen derived some interesting identities on weighted q-Euler polynomials and higher-order q-Euler polynomials from the applications of umbral calculus (see (Araci et al. in J. Number Theory 133(10):3348-3361, 2013)). In this paper, we develop the new method of q-umbral calculus due to Roman, and we study a new q-extension of Euler numbers and polynomials which are derived from q-umbral calculus. Finally, we give some interesting identities on our q-Euler polynomials related to the q-Bernoulli numbers and polynomials of Hegazi and Mansour.

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

A further investigation for Carlitz?s q-Bernoulli numbers and polynomials is performed, and several new formulae for these numbers and polynomials are established by applying some summation transform techniques. Special cases as well as immediate consequences of the main results are also presented.

On the modified multiple q-Bernoulli polynomials with weight

In this paper, we construct new q-extension of Bernoulli polynomials with weight α and weak weight β, and multiple Bernoulli polynomials with weight α and weak weights β1 ,... , βk. These q-Bernoulli polynomials are useful to study various identities of Carlitz’s q-Bernoulli numbers.

New extension of q-Bernoulli polynomials

In this paper, we construct new q-extension of Bernoulli polynomials and Bernoulli polynomials of order k . These q-Bernoulli polynomials are useful to study various identities of Carlitz’s q-Bernoulli numbers.

New extension of q-Bernoulli polynomials with weight

In this paper, we defined a new q-extension of Bernoulli polynomials with weight by generating function, and investigate several properties of those polynomials. These q-Bernoulli polynomials are useful to study various identities of Carlitz’s qBernoulli numbers.

Q-Beta Polynomials and their Applications

The aim of this paper is to construct generating functions for q-beta polynomials. By using these generating functions, we define the q -beta polynomials and also derive some fundamental properties of these polynomials. We give some functional equations and partial differential equations (PDEs) related to these generating functions. By using these equations, we find some identities related to these polynomials, binomial coefficients, the gamma function and the beta function. We obtain a relation between the q- beta polynomials and the q-Bernstein basis functions. We give relations between the q-Beta polynomials, the Bernoulli polynomials, the Euler polynomials and the Stirling numbers. We also give a probability density function associated with the beta polynomials. By applying the Mellin transform, the Fourier transform and the Laplace transform to the generating functions, we obtain not only interpolation function, but also some series representations for the q-Beta polynomials. Furthermore, by using the p-adic q-Volkenborn integral, we give relations between the q-beta polynomials, the q-Euler numbers and the Carlitz's q-Bernoulli numbers.

Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials

In this paper, a further investigation for the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials is performed, and several symmetric identities for these numbers and polynomials are established by applying elementary methods and techniques. It turns out that various known results are derived as special cases.MSC:11B68, 05A19.

Extended p-adic q-invariant integrals on "Equation missing" associated with applications of umbral calculus

Abstract The fundamental aim of this paper is to consider some applications of umbral calculus by utilizing from the extended p-adic q-invariant integral on Z p . From those considerations, we derive some new interesting properties on the extended p-adic q-Bernoulli numbers and polynomials. That is, a systemic study of the class of Sheffer sequences in connection with generating function of the p-adic q-Bernoulli polynomials are given in the present work. MSC:05A10, 11B65, 11B68, 11B73.

Q-Dirichlet type L-functions with weight α

The aim of this paper is to construct q-Dirichlet type L-functions with weight α. We give the values of these functions at negative integers. These values are related to the generalized q-Bernoulli numbers with weight α.AMS Subject Classification:11B68, 11S40, 11S80, 26C05, 30B40.

Generalized q-Stirling Numbers and Their Interpolation Functions

In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions, we construct interpolation functions of these numbers at negative integers. We also derive some identities and relations related to q-Bernoulli numbers and polynomials and q-Stirling numbers of the second kind.

A note on the new approach to q-Bernoulli polynomials

In this paper, we construct new q-extension of Bernoulli poly- nomials. These q-Bernoulli polynomials are useful to study various identities of Carlitz's q-Bernoulli numbers.

Some identities on the twisted q-Bernoulli numbers and polynomials with weight α

In this paper, we consider the twisted q-Bernoulli numbers and polynomials with weight α by using the bosonic q-integral on . From the construction of the twisted q-Bernoulli numbers with weight α, we derive some identities and relations. MSC: 11B68, 11S40, 11S80.

A NOTE ON THE WEIGHTED q-GENOCCHI NUMBERS AND POLYNOMIALS WITH THEIR INTERPOLATION FUNCTION

Recently, T. Kim has introduced and analysed the q-Bernoulli numbers and polynomials with weight <TEX>${\alpha}$</TEX> cf.[7]. By the same motivaton, we also give some interesting properties of the q-Genocchi numbers and polynomials with weight <TEX>${\alpha}$</TEX>. Also, we derive the q-extensions of zeta type functions with weight from the Mellin transformation of this generating function which interpolates the q-Genocchi polynomials with weight at negative integers.