Abstract

In this paper, we consider the degenerate Cauchy numbers of the second kind were defined by Kim (2015). By using modified polyexponential functions, first introduced by Kim-Kim (2019), we define the degenerate poly-Cauchy polynomials and numbers of the second kind and investigate some identities and relationship between various polynomials and the degenerate poly-Cauchy polynomials of the second kind. Using this as a basis of further research, we define the degenerate unipoly-Cauchy polynomials of the second kind and illustrate their important identities.

Highlights

  • We first introduce the Cauchy polynomials Cn ( x ) derived from the integral as follows: Z 1 (1 + t) x+y dy = ∞t tn (1 + t) x = ∑ Cn ( x ), log(1 + t) n! n =0 (1)When x = 0, Cn = Cn (0) are called the Cauchy numbers

  • We consider the degenerate Cauchy numbers of the second kind were defined by Kim

  • In the same motivation of type 2 poly-Bernoulli polynomials arising from modified polyexponential function, we define the degenerate poly-Cauchy polynomials of the second kind as follows: Ei (log(1 + t))

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Summary

Introduction

We first introduce the Cauchy polynomials Cn ( x ) (or the Bernoulli polynomials of the second kind) derived from the integral as follows ( see References [1,2,3,4] ):. Cn,λ,2 ( x ) of the second kind are introduced by Kim to be (see References [11,12,13] ): t log 1 +. When x = 0, Cn,λ,2 = Cn,λ,2 (0) are called the degenerate Cauchy polynomials of the second kind. Pyo-Kim-Kim defined the degenerate Cauchy polynomials Cn,λ,3 ( x ) of the third kind as follows (see References [11,12,13]):. Defined the degenerate Cauchy polynomials Cn,λ,4 ( x ) of the fourth kind as follows We consider the degenerate Cauchy numbers of the second kind were defined by Kim (2015). Using this as a basis of further research, we define the degenerate unipoly-Cauchy polynomials of the second kind and illustrate their important identities

The Degenerate Poly-Cauchy Polynomials of the Second Kind
The Degenerate Unipoly-Cauchy Polynomials of the Second Kind
Conclusions

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