The main goal of the present paper is to construct some families of the Carlitz's $q$-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's $q$-Bernoulli polynomials and numbers with weight ($_{p}$. We then define the modified degenerate Carlitz's $q$-Bernoulli polynomials and numbers with weight ($alpha ,beta $) and obtain some recurrence relations and other identities...

Abstract Recently, B. A. Kupershmidt have constructed a reflection symmetries of q-Bernoulli polynomials (see [9]). In this paper we give another construction of a q-Bernoulli polynomials, which form Barnes’ multiple Bernoulli polynomials at q = 1, cf. [1, 13, 14]. By using q-Volkenborn integration, we can also investigate the properties of the reflection symmetries of these’ q-Bernoulli polyno...

Kupershmidt and Tuenter have introduced reflection symmetries for the q-Bernoulli numbers and the Bernoulli polynomials in 2005 , 2001 , respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the cl...

Abstract. The study of the identities of symmetry for the Bernoulli polynomials arises from the study of Gauss’s multiplication formula for the gamma function. There are many works in this direction. In the sense of p-adic analysis, the q-Bernoulli polynomials are natural extensions of the Bernoulli and Apostol-Bernoulli polynomials (see the introduction of this paper). By using the N-fold iter...

and Applied Analysis 3 we derive some interesting identities and relations on the modified q-Bernoulli numbers and polynomials. 2. The Modified q-Bernoulli Numbers and Polynomials with Weight α In this section, we assume α ∈ Q. Now, we define the modified q-Bernoulli numbers with weight α B̃ α n,q as follows:

and Applied Analysis 3 The purpose of this paper is to derive a new concept of higher-order q-Bernoulli numbers and polynomials with weight α from the fermionic p-adic q-integral on Zp. Finally, we present a systemic study of some families of higher-order q-Bernoulli numbers and polynomials with weight α. 2. Higher Order q-Bernoulli Numbers with Weight α Let β ∈ Z and α ∈ N in this paper. For k...

In this paper, by using the p-adic q-integral on Zp which was defined by Kim, we define the weighted Carlitz q-Bernoulli polynomials and investigate some identities of these polynomials. In particular, we define the weighted degenerate Carlitz’s q-Bernoulli polynomials and numbers and give some interesting properties that are associated with these numbers and polynomials. AMS subject classifica...

and Applied Analysis 3 where n, k ∈ Z see 1, 9, 10 . For n, k ∈ Z , the p-adic Bernstein polynomials of degree n are defined by Bk,n x k x k 1 − x n−k for x ∈ Zp, see 1, 10, 11 . In this paper, we consider Bernstein polynomials to express the p-adic q-integral on Zp and investigate some interesting identities of Bernstein polynomials associated with the q-Bernoulli numbers and polynomials with ...

In this article we prove some relations between two-variable q-Bernoulli polynomials and two-variable q-Euler polynomials. By using the equality eq (z)Eq (−z) = 1, we give an identity for the two-variable qGenocchi polynomials. Also, we obtain an identity for the two-variable q-Bernoulli polynomials. Furthermore, we prove two theorems which are analogues of the q-extension Srivastava-Pinter add...