A NOTE ON TYPE 2 DEGENERATE MULTI-POLY-BERNOULLI POLYNOMIALS AND NUMBERS

Inspired by the denition of degenerate multi-poly-Genocchi polynomials given by using the degenerate multi-polyexponential functions. In this paper, we consider a class of new generating function for the degenerate multi-poly-Bernoulli polynomials, called the type 2 degenerate multi-poly-Bernoulli polynomials by means of the degenerate multiple polyexponential functions. Then, we investigate their some properties and relations. We show that the type 2 degenerate multi-poly-Bernoulli polynomials equals a linear combination of the weighted degenerate Bernoulli polynomials and 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-Bernoulli numbers and degenerate Whitney numbers. 1. Introduction Special polynomials 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. Intense research activities in such an area as the theory of special polynomials are principally motivated by their importance in not only pure and applied mathematics but also mentioned other disciplines. In 1956, Carlitz [1] considered a degenerate form of the well-known Staudt-Clausen theorem and then, in 1979, gave the degenerate versions of the Bernoulli, Stirling and Eulerian numbers in [2]. In spite of their being already more than sixth years old, these studies are still hot topic and today enveloped in an aura of mystery within the scientic community. From 1956 to the present, the degenerate versions of many polynomials, theorems, numbers etc. are investigated intensively by many mathematicians. In fact, in the most recent years, many researchers have been worked on degenerate type various numbers and polynomials. For example, the degenerate Hermite polynomials in [3], the degenerate Hermite-Bernoulli poynomials and numbers in [4], the degenerate Frobenius-Euler-Hermite polynomials in [5], the degenerate Hermite-polyBernoulli polynomials in [6], the degenerate Stirling polynmials of the second kind in [11], the degenerate Bernoulli polynomials of the second kind in [12] and the degenerate Bell polynomials in [13] and degenerate Genocchi polynomials in [19] have been considered and studied extensively. In this paper, we introduce a novel class of degenerate multi-poly-Bernoulli 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-Bernoulli polynomials and numbers and investigate their diverse properties and relations. The last section outlines nding gains and the conclusions in this work and mentions recommentations for future studies. 1991 Mathematics Subject Classication. Primary 11B73, Secondary 11B83, 05A19.


Introduction
Special polynomials 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. Intense research activities in such an area as the theory of special polynomials are principally motivated by their importance in not only pure and applied mathematics but also mentioned other disciplines.
In 1956, Carlitz [1] considered a degenerate form of the well-known Staudt-Clausen theorem and then, in 1979, gave the degenerate versions of the Bernoulli, Stirling and Eulerian numbers in [2]. In spite of their being already more than sixth years old, these studies are still hot topic and today enveloped in an aura of mystery within the scienti…c community. From 1956 to the present, the degenerate versions of many polynomials, theorems, numbers etc. are investigated intensively by many mathematicians. In fact, in the most recent years, many researchers have been worked on degenerate type various numbers and polynomials. For example, the degenerate Hermite polynomials in [3], the degenerate Hermite-Bernoulli poynomials and numbers in [4], the degenerate Frobenius-Euler-Hermite polynomials in [5], the degenerate Hermite-poly-Bernoulli polynomials in [6], the degenerate Stirling polynmials of the second kind in [11], the degenerate Bernoulli polynomials of the second kind in [12] and the degenerate Bell polynomials in [13] and degenerate Genocchi polynomials in [19] have been considered and studied extensively.
In this paper, we introduce a novel class of degenerate multi-poly-Bernoulli 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-Bernoulli polynomials and numbers and investigate their diverse properties and relations.
The last section outlines …nding gains and the conclusions in this work and mentions recommentations for future studies.
To invetigate the derivative property of the type 2 degenerate multi-poly-Bernoulli polynomials, we now consider that (x) t n n! 1 ln (1 + t) which provides the following theorem. is valid for k 1 ; k 2 ; ; k r 2 Z and n 0.

Conclusions
Motivated and inspired by the de…nition of the degenerate multi-poly-Genocchi polynomials introduced by Kim et al. [21], in the present paper, we have considered a class of new generating function for the degenerate multi-poly-Bernoulli polynomials, called the type 2 degenerate multi-poly-Bernoulli polynomials, by means of the degenerate multi-polylogarithm function. Then, we have derived some useful relations and properties. We have shown that the type 2 degenerate multi-poly-Bernoulli polynomials equals a linear combination of the degenerate poly-Bernoulli polynomials and Stirling numbers of the …rst and second kind. In a special case, we have investigated a correlation between the type 2 degenerate multi-poly-Bernoulli numbers and degenerate Whitney numbers.
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. Author Contributions: All authors contributed equally to the manuscript and typed, read, and approved …nal manuscript.