Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function

A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct 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
We have witnessed in recent years that many interesting arithmetic and combinatorial results were obtained in studying degenerate versions of some special polynomials and numbers (see [7][8][9][10][11][12][13] and the references therein), which was initiated by Carlitz when he introduced the degenerate Stirling, Bernoulli and Euler numbers in [3]. The studies have been done with various different tools such as combinatorial methods, generating functions, umbral calculus, p-adic analysis, differential equations, special functions, probability theory and analytic number theory. It should be noted that studying degenerate versions can be done not only for polynomials but also for transcendental functions. Indeed, the degenerate gamma functions were introduced as a degenerate version of ordinary gamma functions in [9]. The degenerate special polynomials and numbers have potential to find diverse applications in many areas just as 'ordinary' special polynomials and numbers play very important role in science and engineering as well as in mathematics. Indeed, it was shown in [10,11] that the expressions of the probability distributions of appropriate random variables can be represented in terms of both the degenerate λ-Stirling polynomials of the second kind and the r-truncated degenerate λ-Stirling polynomials of the second kind.
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.
It is well known that the Gauss hypergeometric function is given by where a 0 = 1, a k = a(a + 1) · · · (a + k -1), (k ≥ 1). The Euler 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 and n k=0 n k = n! (see [4,5]).
For n, m ≥ 0, we have and x n = n k=0 n k x + k n (see [4,5]).
Replacing t by log λ (1 + t) in (15), we get On the other hand, by (2), we get Therefore, by (17) and (18), we obtain the following theorem.
From (20), we note that Therefore, by comparing the coefficients on both sides of (23), we obtain the following theorem.
Therefore, we obtain the following theorem.

Generalized degenerate Bernoulli polynomials
In this section, we consider the generalized degenerate Bernoulli polynomials which are derived from the Gauss hypergeometric function. In the light of (20), we define the generalized degenerate Bernoulli polynomials by Therefore, by comparing the coefficients on both sides of (36), we obtain the following theorem.
interval. In addition, 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.