New construction of type 2 degenerate central Fubini polynomials with their certain properties

Kim et al. (Proc. Jangjeon Math. Soc. 21(4):589–598, 2018) have studied the central Fubini polynomials associated with central factorial numbers of the second kind. Motivated by their work, we introduce degenerate version of the central Fubini polynomials. We show that these polynomials can be represented by the fermionic p-adic integral on Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{Z}_{p}$\end{document}. From the fermionic p-adic integral equations, we derive some new properties related to degenerate central factorial numbers of the second kind and degenerate Euler numbers of the second kind.

Let λ = 0 be any real numbers. Carlitz [1] introduced the degenerate Bernoulli polynomials by means of the following generating function: When x = 0 in (6), β n (λ) =: β n (0; λ) are the degenerate Bernoulli numbers. It is clear from where B n (x) are the Bernoulli polynomials given by e xt , cf. [1].
The degenerate Euler numbers of the second kind are given by It is obvious that where E * n are the Euler numbers of the second kind given by 2 e t + e -t = ∞ n=0 E * n t n n! , cf. [3].
By (5) and (6), the degenerate Stirling polynomials of the second kind are defined by the generating function where x ∈ R, and k is a nonnegative integer (see [13]). In the case x = 0, S (0) 2,λ (n, k) := S 2,λ (n, k) are the degenerate Stirling numbers of the second kind, cf. [1-4, 6, 7]. Since The central factorial numbers of the second kind, denoted by T(j, k) with the conditions j ≥ 0 and k ≥ 0, are defined by where [11,23]).
Kim-Kim [15] introduced the degenerate central factorial polynomials of the second kind as follows: where k is a nonnegative integer. The case x = 0 yields T λ (j, k) =: T λ (j, k|0) that are the degenerate central factorial numbers of the second kind. This paper is organized as follows. In Sect. 2, we consider the generating function of type 2 degenerate central Fubini polynomials and give some properties of these numbers and polynomials. In Sect. 3, we introduce degenerate central Fubini numbers and polynomials and derive some properties of these polynomials by using p-adic fermionic integrals on Z p . In Sect. 4, we introduce type 2 degenerate central Fubini polynomials of two variables and construct some properties of these polynomials. Also, these polynomials are closely related to degenerate central factorial numbers of the second kind and degenerate Euler numbers of the second kind.

On type degenerate central Fubini polynomials
In this section, we assume that λ = 0 is any real number. We begin with giving type 2 degenerate central Fubini polynomials as follows: . (14) Note that (see [11]).
By (14), one may see that Thus, we state the following theorem.
Theorem 2.1 Let n be a nonnegative integer. Then the following holds: The degenerate ordered Bell numbers are defined by the generating function to be where , cf. [3]. (15) gives
Thus, we state the following theorem.

Theorem 2.2
Let n be a nonnegative integer. Then the following relation holds true: 2,λ (n, k).
Thus we arrive at the following theorem.

Theorem 2.3
Let n be a nonnegative integer. Then the following relation between E * n,λ and T λ (n, 2k) holds true: The following computations based on (14) show that Thus, we obtain the following theorem.

On type 2 degenerate central Fubini polynomials by the fermionic p-adic integral on Z p
In this section, let us assume that λ ∈ C p and t ∈ C p with the condition |λt| p < p -1 p-1 . By (3) and (14), it becomes Thus, we get the following theorem.
Theorem 3.1 Let n be a nonnegative integer. The following symmetric relation holds true: By (2) and (14), we have By (19), we get x e It follows from (20) that Thus we state the following theorem.
Theorem 3.2 For n > 0, we have For n ∈ N, by (21), we get Thus we get the following corollary.

On type 2 degenerate central Fubini polynomials of two variable
In this section, we assume that λ = 0 is any real number. We are now in a position to state the type 2 degenerate central Fubini polynomials of two variable as follows: e xt , which is the generating function of the central Fubini polynomials of two variables; see [18].
Thus, we obtain the following theorem.
Theorem 4.1 Let n be a nonnegative integer. Then the following identity holds: Changing t to e λt -1 λ in (22) gives By (5) and (23), we have the following theorem.
Theorem 4.2 Let n be a nonnegative integer. Then the following identity holds: λ n-k F (C) k,λ (x; y)S 2 (n, k).
Therefore, by (26), we obtain the following theorem.

Conclusion
In the present paper, we have considered type 2 degenerate central Fubini and type 2 degenerate central Fubini polynomials of two variables. We investigated some properties, identities and recurrence relations for these polynomials by making use of generating functions and p-adic fermionic integrals on Z p . In addition, we have obtained some results related to degenerate central factorial numbers of the second kind and degenerate Euler numbers of the second kind.