Abstract
We study the symmetry for the generalized twisted Bernoulli polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Bernoulli polynomials using the symmetric properties for the -adic invariant integral.
Highlights
The generalized twisted Bernoulli numbers attached to χ, Bn,χ,ξ, are defined as Bn,χ,ξ Bn,χ,ξ 0 see 16
The purpose of this paper is to study the symmetry for the generalized twisted Bernoulli polynomials and numbers attached to χ
We see that the generalized twisted Bernoulli polynomials attached to χ are given by χ y ξye x y tdy
Summary
Throughout this paper, the symbols Z, Zp, Qp, and Cp denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N∪{0}. Let νp be the normalized exponential valuation of Cp with |p|p p−νp p p−1. Let UD Zp be the space of uniformly differentiable function on Zp. For f ∈ UD Zp , the p-adic invariant integral on Zp is defined as
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