CONSTRUCTION OF THE TYPE 2 DEGENERATE MULTI-POLY-EULER POLYNOMIALS AND NUMBERS

In this paper, we consider a new class of polynomials which is called the multi-poly-Euler polynomials. Then, we investigate their some properties and relations. We provide that the type 2 degenerate multi-poly-Euler polynomials equals a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the rst kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers. 1. Introduction and Preliminaries Special functions has recently been applied in numerous elds of applied and pure mathematics besides in such other disciplines as physics, economics, statistics, probability theory, biology and engineering, cf. [1-24] and see also the references cited therein. One of the most important families of special functions is the family of special polynomials, cf. [1,3-8,10,21,23,24]. 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 is rstly considered by Leonard Carlitz [2] by dening the degenerate forms of the Bernoulli, Stirling and Eulerian numbers. In spite of their being more than sixty years old, these studies are still hot topic and today enveloped in an aura of mystery within the scientic community. Now, the degenerate forms of special polynomials are considered and studied intensively by several mathematicians, cf. [6-8,14-21,23]. 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 by using the series manipulation method and diverse special proof techniques were derived in [6]. Duran and Sadjang [8] considered the fully degenerate Gould-Hopper polynomials with a q parameter and the Gould-Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and then gave some of their properties including not only di¤erence rule, inversion formula and addition formula but also multifarious correlations and implicit summation formulas. Kim et al. [14] introduced degenerate polyexponential functions and the degenerate type 2 poly-Bernoulli polynomials, and also provided some explicit expressions and various identities. 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] dened a new type of the degenerate poly-Genocchi polynomials and numbers constructed from the modied 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 1991 Mathematics Subject Classication. Primary 11B73, Secondary 11B83, 05A19.


Introduction and Preliminaries
Special functions has recently been applied in numerous …elds 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]. 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 is …rstly considered by Leonard Carlitz [2] by de…ning the degenerate forms of the Bernoulli, Stirling and Eulerian numbers. In spite of their being more than sixty years old, these studies are still hot topic and today enveloped in an aura of mystery within the sci-enti…c community. Now, the degenerate forms of special polynomials are considered and studied intensively by several mathematicians, cf. [6][7][8][14][15][16][17][18][19][20][21]23]. 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 by using the series manipulation method and diverse special proof techniques were derived in [6]. Duran and Sadjang [8] considered the fully degenerate Gould-Hopper polynomials with a q parameter and the Gould-Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and then gave some of their properties including not only di¤erence rule, inversion formula and addition formula but also multifarious correlations and implicit summation formulas. Kim et al. [14] introduced degenerate polyexponential functions and the degenerate type 2 poly-Bernoulli polynomials, and also provided some explicit expressions and various identities. 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] de…ned a new type of the degenerate poly-Genocchi polynomials and numbers constructed from the modi…ed 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 function which is multiple version of the degenerate modi…ed polyexponential function and also, by means of this function, considered the degenerate multi-poly-Genocchi polynomials. Moreover, multifarious explicit expressions and some properties were investigated and studied in [20]. Lee et al. [23] studied a new type of the type 2 poly-Euler polynomials and a new type of the type 2 degenerate poly-Euler polynomials by utilizing the modi…ed polyexponential function. Thereafter, several expressions and identities for these polynomials were shown in [23].
In this paper, we introduce a novel class of degenerate multi-poly-Euler polynomials and numbers by means of the degenerate multi-polyexponential function and studied their main explicit relations and identities. This work is organized as follows: Section 2 includes several known de…nitions 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 …nding gains and the conclusions in this work and mentions recommendations for future studies.
Remark 5. Upon setting r = 1, we acquire which is the derivative formula for the new type degenerate poly-Euler polynomials (3.4).
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 (3.4) and the degenerate Stirling numbers of the second kind (2.8).
Kim [17] introduced the degenerate Whitney numbers which are de…ned by the generating function to be Remark 8. In the special case m = 1 and = 0; the degenerate Whitney numbers W m; (n; k j ) reduce to the the degenerate Stirling numbers S 2; (n; k) of the second kind in (2.8), that is, W 1;0 (n; k j ) := S 2; (n; k).
In the future 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.