Abstract
The purpose of this paper is to give some properties of the modified q‐Bernoulli numbers and polynomials of higher order with weight. In particular, by using the bosonic p‐adic q‐integral on ℤp, we derive new identities of q‐Bernoulli numbers and polynomials with weight.
Highlights
Let p be a fixed odd prime number
From Theorem 2.3, we note that qli→m1Bnk,q,α−1 k − x
In an analogues manner as the previous investigation 7–10, we can define a further generalization of modified q-Bernoulli numbers with weight
Summary
Let p be a fixed odd prime number. Throughout this paper Zp, Qp, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp. For n ∈ Z , let us consider the following modified q-Bernoulli numbers with weight α see 1, 3 : Bnα,q Bnα,q x x n qα q−x dμq x Bnα,q x n n l x n−l qα qαlx l0 see 1, 3 . For k ∈ N and n ∈ Z , by making use of the multivariate p-adic q-integral on Zp, we consider the following modified q-Bernoulli numbers with weight α of order k, Bnk,q,α : Bnk,q,α xk n qα q−x1
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