Some Identities Related with the Higher-order Deformed Degenerate Bernoulli and Euler Polynomials

Lee Chae Jang

doi:10.11648/j.acm.20170606.13

Abstract

Recently, Kim-Kim (2016-2017) studied simmetric identities of higher-order degenerate Bernoulli and Euler polynomials which were defined by Carlitz (1979). In this paper, we define the higher-order deformed degenerate Bernoulli and Euler polynomials which are modified the higher-order degenerate Bernoulli and Euler polynomials. We also investigate some interesting identities for the the higher-order deformed degenerate Bernoulli and Euler polynomials.

Highlights

Let be a odd prime number with of ≡ 1 ( 1)
For 1 ∈ N, Carlitz [1, 2] studied the higher-order degenerate Bernoulli polynomials and the higher-order degenerate Euler polynomials which are defined by the generating functions, respectively a
For K, J ∈ C with |J| < nN and |K| < nN, we define the deformed degenerate Bernoulli polynomials which are given by the generating functions to be

Summary

Introduction

Let be a odd prime number with of ≡ 1 ( 1). Throughout this paper, , , and denote the ring of. 1 ∈ N, we consider the higher-order Bernoulli polynomials which are given by the generating function as. For K ≠ 0 , the degenerate Bernoulli polynomials are defined by Carlitz as the generating function to be. LimM→"K :Q:(K, K ) = R:( ) , where R:( ) are the Bernoulli polynomials of the second kind given by the generating function. The degenerate Euler polynomials are defined by Carlitz as the generating function to be. For 1 ∈ N, Carlitz [1, 2] studied the higher-order degenerate Bernoulli polynomials and the higher-order degenerate Euler polynomials which are defined by the generating functions, respectively a. J: , _)i D!, where R: are the higher-order Bernoulli numbers of second kind which are given by the generating functions to be. We investigate some interesting identities for the the higher-order deformed degenerate Bernoulli and Euler polynomials

The Higher-Order Deformed Degenerate Bernoulli and Euler Polynomials

Conclusions