A note on type 2 degenerate poly-Cauchy polynomials and numbers of the second kind

Abstract. Kim-Kim [] studied the type 2 degenerate poly-Bernoulli numbers and polynomials by using modified degenerate polylogarithm function. In this paper, we construct the type 2 degenerate poly-Cauchy polynomials and numbers of the second kind, called degenerate poly-Cauchy polynomials and numbers of the second kind by using degenerate polylogarithm function and derive several properties on the degenerate poly-Cauchy polynomials and numbers of the second kind. Furthermore, we consider the degenerate unipoly-Cauchy polynomials of the second kind and discuss some properties of them.


Introduction
In recent years, many researchers have been studied for various degenerate versions of many special polynomials and numbers which included the degenerate Stirlings numbers of the second kind, degenerate central factorial numbers of the second kind, degenerate Bernoulli numbers of the second kind, degenerate Bernstein polynomials, degenerate Bell numbers and polynomials, degenerate central Bell numbers and polynomials, degenerate complete Bell polynomials and numbers, degenerate Cauchy numbers, and others. (see []). Carlitz [] initiated a study of degenerate versions of some special polynomials and numbers, namely the degenerate Bernoulli and Euler polynomials and numbers.
As is well known, the Cauchy polynomials C n (x) are defined by the means of the following generating function [1,14,15,19] In case when x = 0, C n = C n (0) are called the Cauchy numbers.
For k ∈ Z, the degenerate modified polyexponential function [16] is defined by Kim-Kim to be Note that In [16], Kim-Kim considered the type 2 degenerate poly-Bernoulli polynomials are defined by means of the following generating function In case when n,λ (0) are called the type 2 degenerate poly-Bernoulli numbers.
(1.11) In case when x = 0, C n,λ = C nλ (0) are called the degenerate Cauchy polynomials of the second kind.
The degenerate Stirling numbers of the first kind are defined by [10,16,18]). (1.13) Note here that lim λ→0 S 1,λ (n, k) = S 1 (n, k), where S 1 (n, k) are the Stirling numbers of the first kind given by [7,11]). (1.14) The degenerate Stirling numbers of the second kind are given by , (see [9]). (1.15) Observe here that lim λ→0 S 2,λ (n, k) = S 1 (n, k), where S 2 (n, k) are the Stirling numbers of the second kind given by ). (1.16) By the motivation of the works of Kim-Kim [], we first define the type 2 degenerate poly-Cauchy polynomials of the second kind by using the degenerate polyexponential functions. We investigate some new properties of these numbers and polynomials and derive some new identities and relations between the new type of degenerate poly-Cauchy polynomials of the second kind. Furthermore, we consider the type 2 degenerate unipoly-Cauchy polynomials of the second kind and discuss some identities of them.

Type 2 degenerate poly-Cauchy polynomials of the second kind
For k ∈ Z, by using equations (1.9) and (1.11), we define the type 2 degenerate poly-Cauchy polynomials of the second kind as follows: n,λ (0) are called the type 2 degenerate poly-Cauchy numbers of the second kind. For k = 1, by (2.1), we note that are called the degenerate Cauchy numbers of the second kind.
By comparing the coefficients on both sides of (2.6), we obtain the following theorem.
In [ ], it is well known that the degenerate Bernoulli polynomials of the second kind are defined by From (1.7), we note that Thus, from (2.1) and (2.9), we have Therefore, by (2.10), we obtain the following theorem.
Let k ≥ 1, be an integer. For s ∈ C, we define the function η k,λ (s) as Ei k,λ (log λ (1 + t)) dt Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 7 July 2020 doi:10.20944/preprints202007.0114.v1 The second integral converges absolutely for any s ∈ C and hence, the second term on the right hand side vanishes at non-positive integers. That is, (2.13) On the other hand, for ℜ(s) > 0, the first integral in (2.13) can be written as which defines an entire function of s. Thus, we may include that η k,λ (s) can be continued to an entire function of s.
From (2.1), we note that On the other hand, Therefor, by (2.21) and (2.22), we obtain the following theorem.
Theorem 2.8. For n ≥ 0, we have

Type 2 degenerate unipoly-Cauchy polynomials of the second kind
Let p be any arithmetic function which is a real or complex valued function defined on the set of positive integers N. Kim-Kim [11] defined the unipoly function attached to polynomials p(x) by Moreover, x n n k = Li k (x), (see [5]), (3.2) is the ordinary polylogaritm function.
In this paper, we define the degenerate unipoly function attached to polynomials p(x) as follows: It is worthy to note that is the degenerate modified polyexponential function.
By using (3.3), we define the type 2 degenerate unipoly-Cauchy polynomials of the second kind as follows: In the case when n,λ,p (0) are called the type degenerate unipoly-Cauchy numbers of the second kind. Let us take p(n) = 1 Γλ . Then we have Thus, by (3.6), we have the following theorem.
Theorem 3.1. Let n ≥ 0 and k ∈ Z, and Γn be a Gamma function. Then, we have Therefore, by comparing the coefficients on both sides of (3.8), we obtain the following theorem.
Theorem 3.2. Let n ∈ N and k ∈ Z. Then we have In particular, From (3.5), we observe that ) t n n! . (3.11) From (3.11) , we obtain the following theorem. From (3.5), we observe that ) t n n! .

Conclusions
In 2020 Kim-Kim considered the type degenerate poly-Bernoulli polynomials by making use of the modified polylogarithm functions and Kim [14] introduced the degenerate Cauchy numbers of the second kind. By using these functions and polynomials, we defined the type 2 degenerate poly-Cauchy polynomials of the second kind and obtained some identities of the degenerate poly-Cauchy numbers of the second kind in Theorems 2.1 and 2.2. In particular, we obtained an identity of the degenerate poly-Cauchy polynomials of the second kind in Theorem 2.3. Furthermore, by using the unipoly functions, we defined the degenerate unipoly-Cauchy polynomials of the second kind(Eq. (3.5)) and obtained some properties of the degenerate unipoly-Cauchy numbers of the second kind(Theorems 3.1, and 3.3). Finally, we obtained an identity of the degenerate unipoly-Cauchy polynomials of the second kind in Theorem 3.3 and gave the identity indicating the relationship of the degenerate unipoly-Cauchy numbers of the second kind and the Daehee numbers and degenerate Cauchy numbers of the second kind in Theorem 3.4.
Author Contributions: All authors contributed equally to the manuscript and typed, read, and approved final manuscript.