Abstract
In this paper, we give identities of symmetry for the generalized higher-order qBernoulli polynomials attached to χ which are derived from the symmetric properties of multivariate p-adic invariant integrals on Zp.
Highlights
The higher order Bernoulli polynomials are defined by the generating function to be t et − 1 r ext =
The purpose of this paper is to give identities of symmetry for the generalized q-Bernoulli polynomials of order r which are derived from the symmetric properties of multivariate padic invariant integrals on Zp
By (20) and (22), we obtain the following theorem
Summary
The higher order Bernoulli polynomials are defined by the generating function to be t et − 1 r ext = When x = 0, Bn(r) = Bn(r)(0) is called the n-th Bernoulli number of order r. The generalized Bernoulli polynomials attached to χ are defined by the generating function to be t edt − 1 d−1 a=0 χ(a)eat ext = For r ∈ N, in view of (1), we may consider the generalized higher-order Bernoulli polynomials attached to χ as follows: t d−1 a=0 χ(a)eat edt − 1 r ext
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