Analytical Properties of Degenerate Genocchi Polynomials of the Second Kind and Some of Their Applications

: The main aim of this study is to deﬁne degenerate Genocchi polynomials and numbers of the second kind by using logarithmic functions, and to investigate some of their analytical properties and some applications. For this purpose, many formulas and relations for these polynomials, including some implicit summation formulas, differentiation rules and correlations with the earlier polynomials by utilizing some series manipulation methods, are derived. Additionally, as an ap-plication, the zero values of degenerate Genocchi polynomials and numbers of the second kind are presented in tables and multifarious graphical representations for these zero values are shown.


Introduction
Recently, many mathematicians, particularly Carlitz [1,2], Kim et al. [3], Sharma et al. [4,5], and Khan et al. [6][7][8], have studied and delivered diverse degenerate variations of many unique polynomials and numbers (such as degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate Fubini polynomials, degenerate Stirling numbers of the first and second kind, and so forth). It is noteworthy that studying degenerate variations is not always most effective when limited to polynomials, but also prolonged to transcendental features, like gamma functions. It is likewise terrific that the degenerate umbral calculus is delivered as a degenerate version of the classical umbral calculus. Degenerate versions of special numbers and polynomials have been explored by way of various techniques, such as combinatorial strategies, producing functions, umbral calculus techniques, p-adic analysis, differential equations, unique capabilities, probability principles, and analytic variety ideas. In this paper, we focus on degenerate Genocchi polynomials and numbers of the second kind. The intention of this paper is to introduce a degenerate version of the Genocchi polynomials and numbers of the second type, the so-called degenerate Genocchi polynomials and numbers of the second type, made from the degenerate exponential characteristic. We derive a few express expressions and identities for those numbers and polynomials. Further, we introduce degenerate Genocchi polynomials of the second kind attached to Dirichlet character χ and establish some properties of these polynomials.
Let p be a fixed odd prime number. Throughout this paper, Z p , Q p , and C p will denote the the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Q p , respectively. The p-adic norm | . | p is normalized as | p | p = p −1 = 1 p . Let ∪D(Z p ) be the space of C p -valued uniformly differentiable functions on Z p .
For ω ≥ 0, the Stirling numbers of the first kind are defined by where (ξ) 0 = 1, and (ξ) ω = ξ(ξ − 1) · · · (ξ − ω + 1) (ω ≥ 1). From (12), it is easy to see that For ω ≥ 0, the Stirling numbers of the second kind are defined by From (14), we attain that This article is structured as follows. In Section 2, we consider degenerate Genocchi polynomials of the second kind and derive some basic properties of these polynomials by using different analytical means of their respective generating functions. In Section 3, we introduce degenerate Genocchi polynomials of the second kind attached to Dirichlet character χ and derive some properties of these polynomials.

Degenerate Genocchi Polynomials of the Second Kind
Let λ, z ∈ C p be | λz | p < p − 1 p−1 . Now, we consider the degenerate Genocchi polynomials of the second kind defined by In the case when ξ = 0, G ω,λ (0) = G ω,λ are called the degenerate Genocchi numbers of the second kind.
Proof. Using the definition (8) and (16), we have Comparing the coefficients of z on both sides, we obtain the result.
Proof. By replacing z with 1 λ (e λz − 1) in (16), we get On the other hand, we have In view of (20) and (21), we obtain the result.

Theorem 5.
For ω ≥ 0, we have Proof. From (16), it is shown that Comparing the coefficients of z, we obtain the result (22).

Theorem 6.
For ω ≥ 0 with d ∈ N, we have Proof. From (16), we find Equating the coefficients of z, we get (23).
For r ∈ N, we define the higher-order degenerate Genocchi polynomials of the second kind given by the generating function ω,λ (0) are called the higher-order degenerate Genocchi numbers of the second kind.
It is worth noting that
Proof. By (35), we note that Comparing the coefficients of z, we get (35).
Proof. By applying the difference operator λ to both sides of Equation (32), we get and then we have Therefore, by (40), we obtain (39).

Theorem 14.
Let ω ≥ 0. Then Proof. By applying the derivative operator δ δξ with respect to ξ to both sides of Equation (32), we have By, comparing the coefficients of z ω on both sides, we get the following theorem.

Degenerate Genocchi Polynomials of the Second Kind Attached with Dirichlet Character χ
Here, we introduce degenerate Genocchi polynomials of the second kind attached with Dirichlet character χ and establish some properties of these polynomials by applying the generating function. First, we present the following definition.
Let d ∈ N with d ≡ 1(mod2) and χ be a Dirichlet character with conductor d. We define generalized degenerate Genocchi polynomials of the second kind attached to χ given by the following generating function 2 log(1 + λz) When ξ = 0, G ω,χ,λ = G ω,χ,λ (0) are called the generalized degenerate Genocchi numbers of the second kind attached to χ.

Conclusions
Motivated by [5,13], in this paper, we defined degenerate Genocchi polynomials of the second kind, which turn out to be classical ones in exceptional cases. We have also derived their explicit expressions and some identities involving them. Later, we introduced the higher-order degenerate Genocchi polynomials of the second kind and deduced their explicit expressions and some identities by using the generating functions method, analytical means, and power series expansions. Additionally, we introduced degenerate Genocchi polynomials of the second kind attached to Dirichlet character χ and obtained some properties of these polynomials.