#### Abstract

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.

#### 1. 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 well-known 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 ).

For , the Stirling numbers of the first kind are defined by the following (see [1, 2, 514]):where and . From (1), it is easy to see that (see [4, 1522])

For , the Stirling numbers of the second kind are defined by the following (see [3, 1116]):

From (3), we see that

The Bernoulli , Euler , and Genocchi polynomials are defined by the following (see [1, 6, 7]):respectively.

Let the Changhee polynomials be given by the following (see [2, 13, 14, 16]):

In the case when are called the Changhee numbers.

Moreover, we have the following (see ):

The polyexponential function as an inverse to the polylogarithm function is defined by Kim and Kim  to be

We note that

In 2019, Kim and Kim  introduced the poly-Bernoulli polynomials which are defined by

Letting , are called the poly-Bernoulli numbers.

For , the polylogarithm function is defined by the following (see [5, 21]):

Note that

Lee et al.  introduced the type 2 poly-Euler polynomials which are given by

In the case when , are called the type 2 poly-Euler numbers.

The Daehee polynomials are defined by the following (see [7, 16]):

When , are called the Daehee numbers.

The 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.

#### 2. 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 . At the same time, we give explicit expressions and identities involving polynomials.

For , we define type 2 poly-Changhee polynomials by means of the following exponential generating function (in a suitable neighborhood of ) including the polyexponential function given as follows:

At the point , are called type 2 poly-Changhee numbers.

For , by using (9) and (16), we see thatwhere are called the Changhee polynomials (see equation (7)).

Theorem 1. Let be the nonnegative number and . Then,

Proof. By (16), we haveIn view of (16) and (19), we get (17).

Corollary 1. Let be the nonnegative number. Then,The higher-order Bernoulli polynomials are defined by the following (see ):where are the Bernoulli polynomials of order by

Theorem 2. Let be the nonnegative number. Then,

Proof. From (4), we haveFor , using equations (21) and (24), we findFrom (25), we observe thatIn view of (25) and (26), we complete the theorem.

Corollary 2. Let , then

Theorem 3. For , we have

Proof. Considering (16), we haveSo, the proof is completed.

Theorem 4. Let , then

Proof. Equation (16) can be written asOn the other hand, we haveTherefore, by (31) and (32), we obtain the result.

Theorem 5. For , we have

Proof. By (16), we see thatwhich completes the proof of the theorem.

Theorem 6. Let be the nonnegative number. Then,

Proof. From (16), we note thatBy (16) and (36), we obtain the result.

Theorem 7. Let be the nonnegative number. Then,

Proof. Replacing by in (16), we getTherefore, by (38) and (39), we get the result.

Theorem 8. Let be nonnegative number. Then,

Proof. From (16), we haveThus, by (41) and (42), we complete the proof.

#### 3. Type 2 Unipoly-Changhee Numbers and Polynomials

Recently, Kim and Kim  introduced the unipoly function bywhere is attached to polynomials and is any arithmetic function which is real or complex.

Putting , (43) to get (see )and it is called the polylogarithm function.

By using (7) and (43), we consider the unipoly-Changhee polynomials attached to polynomials by

When , are called the unipoly-Changhee numbers attached to .

Theorem 9. Let be the nonnegative number. Then,

Proof. Taking in (45), we haveThus, we complete the proof.

Theorem 10. For and , we have

Proof. From (45), we haveSo, the proof is completed.

Corollary 3. Let be the nonnegative number. Then,

Theorem 11. Let be the nonnegative number. Then,

Proof. Using (45), we haveSo, we complete the theorem.

Theorem 12. Let be the nonnegative number. Then,

Proof. By (45), we haveTherefore, by (45) and (54), we obtain the result.

#### 4. Conclusion

In the previous sections, we have touched on the problem of recognizing the algebraic structure underlying the poly-Changhee polynomials as given by definition (16). The analysis is aimed at accounting for the wealth of the properties exhibited by these polynomials within the context of the poly-Changhee numbers and polynomials which provide a unifying formalism where the theory of special functions can be framed inherently. Some analogies with the theory of poly-Changhee numbers and polynomials can be recognized and usefully exploited to infer further properties of these polynomials and links with other special functions. Let us stress that the scheme suggested by the following properties of Changhee numbers and polynomials studied in detail by Kim and Kim  can be applied to connect other special functions of relevance in mathematical physics.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

All authors contributed equally to the manuscript and typed, read, and approved final manuscript.