Symmetry Properties of Higher-Order Bernoulli Polynomials

Symmetry Properties of Higher-Order Bernoulli Polynomials

Let p be a fixed 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 algebraic closure of Qp. For x ∈ Cp, we use the notation x q 1 − q / 1 − q . Let UD Zp be the space of uniformly differentiable functions on Zp, and let vp be the normalized exponential valuation of Cp wi...

IDENTITIES OF SYMMETRY FOR THE HIGHER ORDER q-BERNOULLI POLYNOMIALS

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...

Higher Order Bernoulli Polynomials and Newton Polygons

On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order

Let p be a fixed prime number. 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 see 1–24 . Let UD Zp ...

Some Identities of Generalized Higher-order Carlitz q-Bernoulli Polynomials

In this paper, we give some interesting symmetric identities of generalized higher-order Carlitz q-Bernoulli polynomials attached to χ which are derived from symmetry properties of multivariate p-adic q-integral on X.