Construction of Type 2 Poly-Changhee Polynomials and Its Applications

In this paper, we introduce type 2 poly-Changhee polynomials by using the polyexponential function. We derive some explicit expressions and identities for these polynomials, and we also prove some relationships between poly-Changhee polynomials and Stirling numbers of the first and second kind. Also, we introduce the unipoly-Changhee polynomials by employing unipoly function and give multifarious properties. Furthermore, we provide a correlation between the unipoly-Changhee polynomials and the classical Changhee polynomials.


Introduction
Special polynomials and their generating functions have vital roles in several branches of arithmetic, probability, statistics, mathematical physics, and additionally engineering. Since polynomials are appropriate for applying wellknown operations like by-product and integral, polynomials are helpful to check real-world issues within the said areas. As an example, generating functions for special polynomials with their congruousness properties, repetition relations, process formulae, and regular add involving these polynomials are studied in recent years (see [1][2][3][4]).
Moreover, we have the following (see [13]): e polyexponential function as an inverse to the polylogarithm function is defined by Kim and Kim [12] to be We note that In 2019, Kim and Kim [12] introduced the poly-Bernoulli polynomials which are defined by Letting For k ∈ Z, the polylogarithm function is defined by the following (see [5,21]): Note that Lee et al. [22] introduced the type 2 poly-Euler polynomials which are given by In the case when ξ � 0, E (k) j � E (k) j (0) are called the type 2 poly-Euler numbers.
e Daehee polynomials are defined by the following (see [7,16]): When ξ � 0, D j � D j (0) are called the Daehee numbers. e following paper is as follows. In Section 2, we introduce type 2 poly-Changhee polynomials and numbers and derive some identities of these polynomials. We derive some recurrence relations and relationships between Bernoulli number, Euler numbers, and Daehee numbers. In Section 3, we introduce unipoly-Changhee polynomials and investigate some identities of these polynomials.

Type 2 Poly-Changhee Numbers and Polynomials
In this section, we define type 2 poly-Changhee polynomials by using the polyexponential functions and represent the usual Changhee polynomials (more precisely, the values of Changhee polynomials at 1) when k � 1. At the same time, we give explicit expressions and identities involving polynomials.
For k ∈ Z, we define type 2 poly-Changhee polynomials by means of the following exponential generating function (in a suitable neighborhood of z � 0 ) including the polyexponential function given as follows: At the point ξ � 0, Ch (k) j � Ch (k) j (0) are called type 2 poly-Changhee numbers.

Theorem 1.
Let j be the nonnegative number and k ∈ Z. en, Proof. By (16), we have 2 Journal of Mathematics In view of (16) and (19), we get (17).
□ Corollary 1. Let j be the nonnegative number. en, e higher-order Bernoulli polynomials are defined by the following (see [11]): where B (r) j (ξ) are the Bernoulli polynomials of order α by Theorem 2. Let j be the nonnegative number. en, Proof. From (4), we have Ei k−1 (log(1 + 2ξ)).

Journal of Mathematics
Proof. Equation (16) can be written as On the other hand, we have erefore, by (31) and (32), we obtain the result. □ Theorem 5. For j ≥ 0, we have Proof. By (16), we see that which completes the proof of the theorem. □ Theorem 6. Let j be the nonnegative number. en, Proof. From (16), we note that By (16) Proof. Replacing z by e z − 1 in (16), we get erefore, by (38) and (39), we get the result.