Carlitz's q-Bernoulli Research Articles

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.

Symmetry identities for generalized Carlitz's q-Bernoulli polynomials under S4

Symmetry identities for generalized Carlitz's q-Bernoulli polynomials under S4

Some new identities of symmetry for higher-order Carlitz q-Bernoulli polynomials arising from p-adic q-integral on Zp under the symmetric group of degree five

Let p be a fixed prime number. Throughout this paper Zp, Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp. The p-adic norm is normally defined by as |p|p = 1 p . Let q be an indeterminate in Cp such that |1− q|p < p − 1 p−1 . Then the q-analogue of number x is defined as [x]q = 1−qx 1−q . Note that limq→1[x]q = x. Let UF (Zp) be the space of uniformly differentiable functions on Zp. For f ∈ UD(Zp), the p-adic q-integral is defined by Kim to be

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.

A note on identities of symmetry for generalized Carlitz's q-Bernoulli polynomials

A note on identities of symmetry for generalized Carlitz's q-Bernoulli polynomials

A p-adic approach to identities of symmetry for Carlitz's q-Bernoulli polynomials

A p-adic approach to identities of symmetry for Carlitz's q-Bernoulli polynomials

Some symmetric identities of generalized Carlitz's q-Bernoulli polynomials of the first kind

Some symmetric identities of generalized Carlitz's q-Bernoulli polynomials of the first kind

Some identities of generalized higher-order Carlitz q-Bernoulli polynomials

In this paper, we give some interesting symmetric identities of generalized higher-order Carlitz q-Bernoulli polynomials attached to χ which are derived from symmetry properties of multivariate p-adic q-integral on X.

Identities of symmetry for Carlitz q-Bernoulli polynomials arising from q-Volkenborn integrals on Zp under symmetry group S3

Identities of symmetry for Carlitz q-Bernoulli polynomials arising from q-Volkenborn integrals on Zp under symmetry group S3

Some identities of symmetry for the higher-order Carlitz q-Bernoulli polynomials under third dihedral group D_3 arising from q-Volkenborn integral on Z_p

Some identities of symmetry for the higher-order Carlitz q-Bernoulli polynomials under third dihedral group D_3 arising from q-Volkenborn integral on Z_p

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.

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.

A note on Carlitz q-Bernoulli numbers and polynomials

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

Some new identities on the twisted carlitz's q-bernoulli numbers and q-bernstein polynomials

In this paper, we consider the twisted Carlitz's q-Bernoulli numbers using p-adic q-integral on ℤ p . From the construction of the twisted Carlitz's q-Bernoulli numbers, we investigate some properties for the twisted Carlitz's q-Bernoulli numbers. Finally, we give some relations between the twisted Carlitz's q-Bernoulli numbers and q-Bernstein polynomials.

A rooted-treesq-series lifting a one-parameter family of Lie idempotents

We define and study a series indexed by rooted trees and with coefficients in Q(q). We show that it is related to a family of Lie idempotents. We prove that this series is a q-deformation of a more classical series and that some of its coefficients are Carlitz q-Bernoulli numbers.

A note on p-adic Carlitz's q-Bernoulli numbers

In a recent paper I have shown that Carlitz's q-Bernoulli number can be represented as an integral by the q-analogue μq of the ordinary p-adic invariant measure. In the p-adic case, J. Satoh could not determine the generating function of q-Bernoulli numbers. In this paper, we give the generating function of q-Bernoulli numbers in the p-adic case.