UNIFIED DEGENERATE CENTRAL BELL POLYNOMIALS

In this paper, we firstly consider extended degenerate central factorial numbers of the second kind and provide some properties of them. We then introduce unified degenerate central Bell polynomials and numbers and investigate many relations and formulas including summation formula, explicit formula and derivative property. Moreover, we derive several correlations for the fully degenerate central Bell polynomials associated with the degenerate Bernstein polynomials and the degenerate Bernoulli, Euler and Genocchi numbers.


Introduction
The classical Bell polynomials Bel n (x) (called also Touchard polynomials and exponential polynomials) and central Bell polynomials B The Bell polynomials extensively studied by Bell [2] appear as a standard mathematical tool and arise in combinatorial analysis. The familiar Bell polynomials and the central Bell polynomials have been intensely studied by many mathematicians, cf. [2-4, 8, 9, 11-13, 16, 17, 20-26] and see also the references cited therein. For example, Bouroubi [3] provides a novel and interesting approach to the determination of new formulas for the Bell polynomials, based on the Lagrange inversion formula, and the binomial sequences which gives the easy recovery of known relations and deduction of several new formulas covering these polynomials. Carlitz [4] investigate diverse formulas for the Bell numbers including correlations with the Stirling numbers of the second kind, combinatorial interpretation and derivative property. Kim et al. [8] considered the central Bell polynomials and numbers and presented several relations and identities for these polynomials and numbers. Kim et al. [9] analyses properties of the Bell polynomials by using and without using umbral calculus and proved several representations for multifarious known families of polynomials such as Cauchy polynomials, Bernoulli polynomials, poly-Bernoulli polynomials and falling factorials by means of the Bell polynomials. Mihoubi [20] gave some results tied the Bell polynomials and the binomial type sequences, which were used to derive some novel formulas for the Bell polynomials. Qi [22] derived an explicit formula for the Bell numbers in terms of the Lah numbers and the Stirling numbers of the second kind. Qi [25] pprovided derivatives of the generating functions for the Bell numbers by induction and by the Faà di Bruno formula, recovered an explicit formula in terms of the Stirling numbers of the second kind, derived the (logarithmically) absolute and complete monotonicity of the generating functions, and constructed some inequalities for the Bell numbers. Qi et al. [26] presented two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, obtained two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recovered a known identity connecting the sequence of unnamed polynomials with the Bell polynomials. These large investigations of the Bell polynomials and numbers yields a motivation to improve this mathematical tool. For non-negative integer n, the central factorial numbers of the second kind T (n, m) are defined by the following exponential generating function [8,11,17]) (1.4) or by recurrence relation for a fixed non-negative integer n, where the symbol x [m] called as the central factorial equals to with initial condition x [0] = 1, cf. [8,11,17] and see also references cited therein. The central Bell polynomials and central factorial numbers of the second kind satisfy the following relation (cf. [8,11,17]) The Stirling numbers of the first kind S 1 (n, m) are defined as follows (cf. [7-9, 11-14, 17, 20-28]) where the notation (x) n called as the falling factorial equals to x (x − 1) · · · (x − n + 1), cf. [7-9, 11-14, 17, 20-28] and see also references cited therein.
The following sections are planned as follows: The second section includes the definition of the unified degenerate central Bell polynomials and numbers and provides several formulas and relations including the unified degenerate central factorial numbers of the second kind and Stirling numbers of the first kind. The third section covers diverse correlations for the unified degenerate central Bell polynomials associated with the degenerate Bernstein polynomials, the degenerate Bernoulli, Euler and Genocchi numbers. The last section of this paper analyses the results acquired in this paper.

Unified Degenerate Central Bell Polynomials
In this section, we perform to analyze and investigate degenerate forms of some special polynomials and numbers. We focus on the unified degenerate central factorial numbers of the second kind and the unified degenerate central Bell polynomials and numbers. We then derive several properties and formulas for these polynomials.
In the theory of special polynomials and special functions, the degenerate forms for polynomials and functions have been studied and investigated by many mathematicians for more than a century, cf. [4-6, 9-19, 28-31] and see also the references cited therein. Carlitz [5] introduced higher order degenerate Euler polynomials and provided several properties. Carlitz [6] gave the degenerate Staudt-Clausen theorem and illustrated it for the degenerate Bernoulli numbers. Howard [7] proved some explicit formulas for degenerate Bernoulli polynomials. Kim et al. [10] considered the degenerate Bernstein polynomials and attained their generating function, recurrence relations, symmetric identities, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind. Kim et al. [11] studied on the degenerate central Bell numbers and polynomials and derived some properties, identities, and recurrence relations. Kim et al. [12] considered degenerate Bell numbers and polynomials and presented several novel formulas for those numbers and polynomials associated with special numbers and polynomials by using the notion of composita. Kim et al. [13] acquired diverse properties, recurrence relations, and identities associated with the degenerate r-Stirling numbers of the second kind and the degenerate r-Bell polynomials by means of umbral calculus. Kim et al. [14] presented various explicit formulas and recurrence relationships for the degenerate Mittag-Leffler polynomials and also gave several connections between Mittag-Leffler polynomials and other known families of polynomials. Kim et al. [15] introduced the degenerate Laplace transform and degenerate gamma function and obtained several interesting formulas related to this transform and this gamma function. Kim et al. [16] considered partially degenerate Bell polynomials numbers and developed their properties and identities. Kim et al. [17] defined and studied on the extended degenerate rcentral factorial numbers of the second kind and the extended degenerate r-central Bell polynomials. Kwon et al. [18] considered degenerate Changhee polynomials and proved several relations and formulas for these polynomials. Lim [19] defined higher order degenerate Genocchi polynomials and gave some identities and formulas for these polynomials. Qi et al. [31] defined partially degenerate Bernoulli polynomials of the first kind and investigated several properties.
For non-negative integer n, the degenerate central factorial numbers of the second kind T 2,λ (n, m) are defined by the following exponential generating function where the notation e x λ (t) denotes the degenerate exponential function for a real number λ, given by It is readily seen that lim λ→0 e x λ (t) = e xt , cf. [11] and [17]. Remark. When λ approaches to 0, the degenerate central factorial numbers of the second kind (2.1) reduces to the central factorial numbers of the second kind (1.4), namely lim λ→0 T 2,λ (n, m) = T (n, m).
We are now ready to give the definition of the unified degenerate central factorial numbers of the second kind.
Definition 2.1. Let λ and ω be real numbers. The unified degenerate central factorial numbers of the second kind T 2,λ;ω (n, m) are introduced by means of the following generating function We here analyze some circumstances of the unified degenerate central factorial numbers of the second kind T 2,λ;ω (n, m) as follows.
(2) When λ → 0, the unified degenerate central factorial numbers of the second kind T 2,λ;ω (n, m) reduce to the ω-analogue of the central factorial numbers of the second kind denoted by T 2;ω (n, m), which is also new generalization of the factorial numbers of the second kind T (n, m) in (1.4), given by (3) When ω = 1 2 and λ → 0, we obtain the usual central factorial numbers of the second kind T (n, m) in (1.4), cf. [8,11,17].
We now investigate some properties of the unified degenerate central factorial numbers of the second kind T 2,λ;ω (n, m). Hence, we give the following Theorem 2.2.
Theorem 2.2. For non-negative integers k and m, we have Proof. In view of Definition 2.1, we write and then we get which implies the asserted result (2.5).
We here give the following correlation.
Theorem 2.3. The following correlation is valid for real numbers λ and ω.
Proof. By Definition 2.1 and the identity (2.2), we obtain which provides the desired result (2.6).
The degenerate classical Bell polynomials and the degenerate central Bell polynomials are given by the following Taylor series expansion at t = 0 as follows: For non-negative integer n and a real number λ, λ-extension of falling factorial is defined by (cf. [10,16]) and can be expressed by the usual falling factorial as follows: (x) n,λ = λ n x λ n n ≥ 0.
From (2.2) and (2.11), we obtain the following relation (cf. [11,17]) The degenerate central Bell polynomials and the degenerate central factorial numbers of the second kind satisfy the following relation (cf. [11]) and The immediate relation for the unified degenerate central Bell polynomials and numbers is B n,λ,α (ω). We now examine some special cases of the unified degenerate central Bell polynomials as follows.

Remark.
(1) When ω = 1 2 , the unified degenerate central Bell polynomials B  n,λ,α (x : ω). Thus, we firstly give the following theorem that includes a formula which is the generalization of the relations (1.6) and (2.13). holds true for real numbers α, λ and ω. is valid for real numbers α, λ and ω.

Proof. By Definition 2.4 and formulas (2.2) and (2.12), we get
Proof. By Definition 2.4 and the identity (2.12), we obtain which provides the desired result (2.20).
We now provide a correlation as follows.
We give the following theorem.
which implies the desired result (2.23).
We now present a derivative property for B (c) n,λ,α (x : ω) as follows.

Connections with Some Known Degenerate Polynomials and Numbers
The main aim of this section is to derive diverse connections with some earlier degenerate polynomials such as Bernstein, Bernoulli, Genocchi and Euler polynomials for the unified degenerate central Bell polynomials. Thanks to this purpose, we acquire multifarious correlations in the family of the degenerate polynomials.
We firstly perform to obtain a relation with the degenerate Bernstein polynomials.
We start with the following computations where the degenerate Bernstein polynomials are defined by cf. [10]. Thus, we obtain the following theorem.
Theorem 3.1. The following relation is valid.
Theorem 3.2. The following summation correlation In the special cases of the Theorems 3.1 and 3.2, we get new formulas for the usual central Bell polynomials and familiar Bernstein polynomials as follows.
A relation including the degenerate Euler numbers and the unified degenerate central Bell polynomials is given by the following theorem. A relationship for the degenerate Genocchi numbers and the unified degenerate central Bell polynomials is stated in the following theorem. n+1−k,λ,α (x : ω) G k,λ 2 (n + 1) holds true.

Conclusion
In this paper, we have firstly generalized the central factorial function termed as extended degenerate central factorial numbers of the second kind and have given some identities for the mentioned numbers. We then have defined unification of the degenerate central Bell polynomials and numbers. We have analyzed multifarious properties and formulas for the aforesaid polynomials and numbers. We have provided several correlations for the unified degenerate central polynomials related to the degenerate Bernstein polynomials and the degenerate Bernoulli, Euler and Genocchi numbers.