Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function

In a previous paper, Rahmani introduced a new family of p-Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. Motivated by this paper and as a degenerate version of those numbers and polynomials, we introduce the generalized degenerate Bernoulli numbers and polynomials again by using the Gauss hypergeometric function. In addition, we introduce the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate $p$-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.


INTRODUCTION
As the first degenerate versions of some special numbers, Carlitz introduced the degenerate Stirling, Bernoulli and Euler numbers in [3]. In recent years, degenerate versions of many special polynomials and numbers have been investigated by means of various different tools including generating functions, combinatorial methods, umbral calculus, p-adic analysis, differential equations, special functions, probability theory and analytic number theory. Here we would like to remark that studying degenerate versions of some special polynomials and numbers has yielded many interesting arithmetic and combinatorial results (see [7-13 and references therein]) and has potential to find many applications to diverse areas in science and engineering as well as in mathematics. For example, it was shown in [10,11] that both the degenerate λ -Stirling polynomials of the second kind and the r-truncated degenerate λ -Stirling polynomials of the second kind appear in the expressions of the probability distributions of appropriate random variables. Also, we would like to emphasize that studying degenerate versions is applied not only to polynomials but also to transcendental functions. Indeed, the degenerate gamma functions were introduced and some interesting results were derived in [9].
In [14], Rahmani introduced a new family of p-Bernoulli numbers and polynomials by means of the Gauss hypergeometric function which reduce to the classical Bernoulli numbers and polynomials for p = 0. Motivated by that paper and as a degenerate version of those numbers and polynomials, in this paper we introduce the generalized degenerate Bernoulli numbers and polynomials again in terms of the Gauss hypergeometric function which reduce to the Carlitz degenerate Bernoulli numbers and polynomials for p = 0. In addition, we introduce the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. The aim of this paper is to study the generalized degenerate Bernoulli numbers and polynomials and to show their connections to other special numbers and polynomials. Among other things, for the generalized degenerate Bernoulli numbers we express them in terms of the degenerate Stirling numbers of the second kind , of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind. For the rest of this section, we recall the necessary facts that are needed throughout this paper.
In [7], the degenerate Stirling numbers of the first kind are defined by As the inversion formula of (3), the degenerate Stirling numbers of the second kind are defined by [7]).
It is well known that the Gauss hypergeometric function is given by The Pfaff's transformation formula is given by and the Euler's transformation formula is given by The Eulerian number n k is the number of permutation {1, 2, 3, . . . , n} having k permutation ascents. The Eulerian numbers are given explicitly by the finite sum n ∑ k=0 n k = n!, (see [4,5]).
Recently, the degenerate Stirling polynomials of the second kind are defined by Thus, by (14), we get  [3]).
Theorem 1. For n ≥ 0, we have Replacing t by log λ (1 + t) in (16), we get On the other hand, by (2), we get Therefore, by (18) and (19), we obtain the following theorem.
From (16) and (17), we note that In view of (20), we may consider the generalized degenerate Bernoulli numbers given in terms of Gauss hypergeometric function by n,λ = β n,λ , (n ≥ 0). Let us take p = −1 in (21). Then we have By comparing the coefficients on the both sides of (22), we get n,λ = (λ − 1) n,λ , (n ≥ 0). From (21), we note that Therefore, by comparing the coefficients on both sides of (24), we obtain the following theorem.
By (25), we get From (26), we have In light of (12), we may consider the degenerate type Eulerian numbers given by By (27) and (28), we get We observe that From (30) and Theorem 3, we note that Therefore, by (31), we obtain the following theorem. into k non-empty disjoint subsets in such a way that the numbers 1, 2, 3, . . . , r are in distinct subsets. In [13], Kim-Kim-Lee-Park introduced the unsigned degenerate r-Stirling numbers of the first kind n k r,λ as a degenerate version of n k r and the degenerate r-Stirling number of the second kind n k r,λ as a degenerate version of n k r . It is known that the degenerate r-Stirling numbers of the second kind are given by From (32), we note that (33) 1 k! e λ (t) − 1 k e r λ (t) = ∞ ∑ n=k n + r k + r r,λ t n n! , (k ≥ 0, r ≥ 1).
Theorem 5. For n ≥ 1 and p ≥ 0, we have Note that From Theorem 3, we have Therefore, we obtain the following theorem.

CONCLUSION
In recent years, degenerate versions of many special polynomials and numbers have been investigated by means of various different tools including generating functions, combinatorial methods, umbral calculus, p-adic analysis, differential equations, special functions, probability theory and analytic number theory. Studying degenerate versions of some special polynomials and numbers, which was initiated by Carlitz in [3], has yielded many interesting arithmetic and combinatorial results (see [7-13 and references therein]) and has potential to find many applications in diverse areas.
A new family of p-Bernoulli numbers and polynomials, which reduce to the classical Bernoulli numbers and polynomials for p = 0, was introduced by Rahmani in [14] by means of the Gauss hypergeometric function. Motivated by that paper, we were interested in finding a degenerate version of those numbers and polynomials. Indeed, the generalized degenerate Bernoulli numbers and polynomials, which reduce to the Carlitz degenerate Bernoulli numbers and polynomials for p = 0, were introduced again in terms of the Gauss hypergeometric function. In addition, the degenerate type Eulerian numbers was introduced as a degenerate version of Eulerian numbers.
In this paper, we expressed the generalized degenerate Bernoulli numbers in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. In dddition, we represented the generalized degenerate Bernoulli polynomials in terms of the degenerate Stirling polynomials of the second kind.
It is one of our future projects to continue pursuing this line of research. Namely, by studying degenerate versions of some special polynomials and numbers, we want to find their applications in mathematics, science and engineering.