Differential equations arising from the generating function of general modified degenerate Euler numbers

Tae Kyun Kim,Jong Jin Seo,Dae San Kim,Hyuck In Kwon

doi:10.1186/s13662-016-0858-7

Abstract

In this paper, we introduce the general modified degenerate Euler numbers and study ordinary differential equations arising from the generating function of these numbers. In addition, we give some new explicit identities for the general modified degenerate Euler numbers arising from our differential equations.

Highlights

As is known, the Euler numbers are defined by the generating function∞ tn et + = En n!. ( . ) n=Carlitz [ ] considered the degenerate Euler numbers defined by the generating function ∞ tn ( + λt) λ + = En,λ n! .In [ ], the modified degenerate Euler numbers, which are slightly different from Carlitz’s degenerate Euler numbers, are defined by t
Kim and Kim [ ] studied nonlinear differential equations given by d dt
Bayad and Kim [ ] studied the following nonlinear differential equations related to Apostol-Euler numbers: FqN

Summary

Introduction

Introduction As is known, the Euler numbers are defined by the generating function Carlitz [ ] considered the degenerate Euler numbers defined by the generating function Kim and Kim [ ] studied nonlinear differential equations given by d dt Let α, a, b be nonzero real numbers. We consider the general modified degenerate Euler numbers as follows:

Results

Conclusion