The modified degenerate q-Bernoulli polynomials arising from p-adic invariant integral on mathbb{Z}_{p}

Jeong Gon Lee,Jongkyum Kwon

doi:10.1186/s13662-017-1084-7

Abstract

Dolgy et al. introduced the modified degenerate Bernoulli polynomials, which are different from Carlitz’s degenerate Bernoulli polynomials (see Dolgy et al. in Adv. Stud. Contemp. Math. (Kyungshang) 26(1):1-9, 2016). In this paper, we study some explicit identities and properties for the modified degenerate q-Bernoulli polynomials arising from the p-adic invariant integral on mathbb{Z}_{p}.

Highlights

For a fixed prime number p, Zp refers to the ring of p-adic integers, Qp to the field of p-adic rational numbers, and Cp to the completion of algebraic closure of Qp
The modified degenerate Bernoulli polynomials are recently revisited by Dolgy et al, and they are formulated with the p-adic invariant integral on Zp to be
The generating functions of Stirling numbers are given by log( + t) tl S (l, n) l!

Summary

Introduction

For a fixed prime number p, Zp refers to the ring of p-adic integers, Qp to the field of p-adic rational numbers, and Cp to the completion of algebraic closure of Qp. The Bernoulli polynomials are given by the generating function t et – Carlitz [ , , ] defined the degenerate Bernoulli polynomials as follows: t When x = , βn( |λ) = βn(λ) are called Carlitz’s degenerate Bernoulli numbers. The modified degenerate Bernoulli polynomials are recently revisited by Dolgy et al, and they are formulated with the p-adic invariant integral on Zp to be

Results

Conclusion