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Note on the Type 2 Degenerate Multi-Poly-Euler Polynomials

Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown.


Introduction and Preliminaries
Special functions have recently been applied in numerous fields of applied and pure mathematics besides in such other disciplines as physics, economics, statistics, probability theory, biology and engineering, cf. , and see also the references cited therein. One of the most important families of special functions is the family of special polynomials, cf. [1,[3][4][5][6][7][8]10,21,[23][24][25][26]. Intense research activities in such an area as the family of special polynomials are principally motivated by their importance in both pure and applied mathematics and other disciplines. The degenerate forms of special polynomials are firstly considered by Leonard Carlitz [2] by defining the degenerate forms of the Bernoulli, Stirling, and Eulerian numbers. Despite there being more than 60 years old, these studies are still a hot topic and today enveloped in an aura of mystery within the scientific community, cf. [6][7][8][14][15][16][17][18][19][20][21]23,25,26]. For instance, Duran and Acikgoz [6] considered the degenerate truncated exponential polynomials and gave their several properties. After that, degenerate truncated forms of various special polynomials including Genocchi, Bell, Bernstein, Fubini, Euler, and Bernoulli polynomials were introduced via the degenerate truncated exponential polynomials and their various properties and relationships were derived in [6]. Kim and Kim [15] considered degenerate poly-Bernoulli polynomials by means of the degenerate polylogarithm function and investigated several properties and relations. Kim et al. [16] defined a new type of the degenerate poly-Genocchi polynomials and numbers constructed from the modified polyexponential function and the degenerate unipoly Genocchi polynomials and derived several combinatorial identities and some explicit expressions. Kim [17] introduced a degenerate form of the Stirling polynomials of the second kind and proved some novel relations and identities for these polynomials. Kim and Kim [18] considered a new type degenerate Bell polynomials via degenerate polyexponential functions and then gave some of their properties. Kim et al. [20] introduced degenerate multiple polyexponential functions whereby the degenerate multi-poly-Genocchi polynomials are considered and multifarious explicit expressions and some properties were investigated. Lee et al. [25] studied a new type of type 2 poly-Euler polynomials and its degenerate form by utilizing the modified polyexponential function.
In this paper, we introduce a novel class of degenerate multi-poly-Euler polynomials and numbers utilizing the degenerate multi-polyexponential function and studied their main explicit relations and identities. This work is organized as follows: • Section 2 includes several known definitions and notations.

•
In Section 3, we consider a novel class of degenerate multi-poly-Euler polynomials and numbers and investigate their diverse properties and relations.

•
The last section outlines finding gains and the conclusions in this work and mentions recommendations for future studies.
In [25], the type 2 poly-Euler polynomials E (k) n (x) and the type 2 degenerate poly-Euler polynomials E (k) n,λ (x) are introduced using the following generating functions to be Multifarious relations and identities for these polynomials are investigated intensely in [25].

Remark 2.
In the case when r = 1, the type 2 degenerate multi-poly-Euler polynomials reduce to a new type degenerate poly-Euler polynomials that we denote E (6) defined by Lee et al. [25], as follows: Also, when x = 0 in (12), these new type degenerate poly-Euler polynomials E Now, we investigate some properties of the type 2 degenerate multi-poly-Euler polynomials.

Theorem 1. The following relation
holds true for n ≥ 0.
Proof. From Definition 1, we observe that which gives the desired result (13).

Remark 3.
When λ approaches to 0, we get the following known relation for the multi-poly-Euler polynomials (cf. [4,10]) The degenerate Euler polynomials of higher order are given by the following Maclaurin series: cf. [1,6,25], and also see the references cited therein. We also notice that when r = 1, the degenerate Euler polynomials of higher order reduce to the degenerate Euler polynomials in (3), namely, A summation formula for the type 2 degenerate multi-poly-Euler polynomials is stated in the following theorem.

Remark 4.
When r = 1, we have which is a new relation including a new type degenerate poly-Euler polynomials (12), degenerate Euler polynomials (3), and degenerate Stirling numbers of the first kind (7).
Proof. Given Definition 1, we see that which implies the claimed result (15).
The derivative property of the type 2 degenerate multi-poly-Euler polynomials is provided below.
Proof. By Definition 1, we observe that which provides the asserted result (16).

Remark 5.
Upon setting r = 1, we acquire which is the derivative formula for the new type degenerate poly-Euler polynomials (12).

Remark 6.
Taking r = k = 1, we attain which is the derivative formula for the degenerate Euler polynomials, cf. [6].

Remark 7.
In the case when r = 1, we acquire which is a relation for the new type degenerate poly-Euler polynomials (12) and the degenerate Stirling numbers of the second kind (8).
Kim [17] introduced the degenerate Whitney numbers which are defined by the generating function to be (e m

Remark 9.
Upon setting r = 1, we get which is a relation between the degenerate Whitney numbers and the new type degenerate poly-Euler polynomials (12).
By means of the multiple polylogarithm function, the degenerate multi-poly-Bernoulli polynomials are introduced (cf. [4,10,19]) as follows Then, several properties for those polynomials are investigated.
Then, we have derived some useful relations and properties. In a special case, we have investigated a correlation including the type 2 degenerate multi-poly-Euler polynomials and numbers, and degenerate Whitney numbers. We have also analyzed several special circumstances of the results derived in this paper.
In the plans, we will continue to study degenerate versions of certain special polynomials and numbers and their applications to probability, physics and engineering in addition to mathematics.
Author Contributions: All authors contributed equally to the manuscript and typed, read and approved the final manuscript. All authors have read and agreed to the published version of the manuscript.