Abstract
We investigate properties of identities and some interesting identities of symmetry for the Bernoulli polynomials of higher order using the multivariate -adic invariant integral on .
Highlights
The purpose of this paper is to investigate some interesting properties of symmetry for the multivariate p-adic invariant integral on Zp
From the properties of symmetry for the multivariate p-adic invariant integral on Zp, we derive some interesting identities of symmetry for the Bernoulli polynomials of higher order
In 2.1, we note that D m w1, w2 is symmetric in w1, w2
Summary
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 algebraic closure of Qp. For x ∈ Cp, we use the notation x q 1 − qx / 1 − q. Let UD Zp be the space of uniformly differentiable functions on Zp, and let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p. For q ∈ Cp with |1 − q|p < 1, the q-Volkenborn integral on Zp is defined as
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