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.

A Note on Type-Two Degenerate Poly-Changhee Polynomials of the Second Kind

In this paper, we first define type-two degenerate poly-Changhee polynomials of the second kind by using modified degenerate polyexponential functions. We derive new identities and relations between type-two degenerate poly-Changhee polynomials of the second kind. Finally, we derive type-two degenerate unipoly-Changhee polynomials of the second kind and discuss some of their identities.

A new class of Appell-type Changhee-Euler polynomials and related properties

<abstract><p>A remarkably large number of polynomials and their extensions have been presented and studied. In the present paper, we introduce the new type of generating function of Appell-type Changhee-Euler polynomials by combining the Appell-type Changhee polynomials and Euler polynomials and the numbers corresponding to these polynomials are also investigated. Certain relations and identities involving these polynomials are established. Further, the differential equations arising from the generating function of the Appell-type Changhee-Euler polynomials are derived. Also, the graphical representations of the zeros of these polynomials are explored for different values of indices.</p></abstract>

Truncated-Exponential-Based Appell-Type Changhee Polynomials

The truncated exponential polynomials em(x) (1), their extensions, and certain newly-introduced polynomials which combine the truncated exponential polynomials with other known polynomials have been investigated and applied in various ways. In this paper, by incorporating the Appell-type Changhee polynomials Chn*(x) (10) and the truncated exponential polynomials in a natural way, we aim to introduce so-called truncated-exponential-based Appell-type Changhee polynomials eCn*(x) in Definition 1. Then, we investigate certain properties and identities for these new polynomials such as explicit representation, addition formulas, recurrence relations, differential and integral formulas, and some related inequalities. We also present some integral inequalities involving these polynomials eCn*(x). Further we discuss zero distributions of these polynomials by observing their graphs drawn by Mathematica. Lastly some open questions are suggested.

Some Relations of Two Type 2 Polynomials and Discrete Harmonic Numbers and Polynomials

Harmonic numbers appear, for example, in many expressions involving Riemann zeta functions. Here, among other things, we introduce and study discrete versions of those numbers, namely the discrete harmonic numbers. The aim of this paper is twofold. The first is to find several relations between the Type 2 higher-order degenerate Euler polynomials and the Type 2 high-order Changhee polynomials in connection with the degenerate Stirling numbers of both kinds and Jindalrae–Stirling numbers of both kinds. The second is to define the discrete harmonic numbers and some related polynomials and numbers, and to derive their explicit expressions and an identity.

Notes on the Poly-Korobov polynomials and related polynomials

In recent years, many mathematicians ([2], [7], [8], [9], [15], [16], [21]) introduced and investigated for the Korobov polynomials. They gave some identities and relations for the Korobov type polynomials. In this work, we give some relations for the first kind Korobov polynomials and Korobov type Changhee polynomials. Further, wegive two relations between the poly-Changhee polynomials and the poly-Korobov polynomials. Also, we give a relation among the poly-Korobov type Changhee polynomials, the Stirling numbers of the second kind, the Euler polynomials and the Bernoulli numbers.

Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

The purpose of this paper is to construct generating functions for negative order Changhee numbers and polynomials. Using these generating functions with their functional equation, we prove computation formulas for combinatorial numbers and polynomials. These formulas include Euler numbers and polynomials of higher order, Stirling numbers, and negative order Changhee numbers and polynomials. We also give some properties of these numbers and polynomials with their generating functions. Moreover, we give relations among Changhee numbers and polynomials of negative order, combinatorial numbers and polynomials and Bernoulli numbers of the second kind. Finally, we give a partial derivative of an equation for generating functions for Changhee numbers and polynomials of negative order. Using these differential equations, we derive recurrence relations, differential and integral formulas for these numbers and polynomials. We also give p-adic integrals representations for negative order Changhee polynomials including Changhee numbers and Deahee numbers.

On applications for Mahler expansion associated with p-adic q-integrals

In this paper, we primarily consider a generalization of the fermionic [Formula: see text]-adic [Formula: see text]-integral on [Formula: see text] including the parameters [Formula: see text] and [Formula: see text] and investigate its some basic properties. By means of the foregoing integral, we introduce two generalizations of [Formula: see text]-Changhee polynomials and numbers as [Formula: see text]-Changhee polynomials and numbers with weight [Formula: see text] and [Formula: see text]-Changhee polynomials and numbers of second kind with weight [Formula: see text]. For the mentioned polynomials, we obtain new and interesting relationships and identities including symmetric relation, recurrence relations and correlations associated with the weighted [Formula: see text]-Euler polynomials, [Formula: see text]-Stirling numbers of the second kind and Stirling numbers of first and second kinds. Then, we discover multifarious relationships among the two types of weighted [Formula: see text]-Changhee polynomials and [Formula: see text]-adic gamma function. Also, we compute the weighted fermionic [Formula: see text]-adic [Formula: see text]-integral of the derivative of [Formula: see text]-adic gamma function. Moreover, we give a novel representation for the [Formula: see text]-adic Euler constant by means of the weighted [Formula: see text]-Changhee polynomials and numbers. We finally provide a quirky explicit formula for [Formula: see text]-adic Euler constant.

Symmetry Identities of Changhee Polynomials of Type Two

In this paper, we consider Changhee polynomials of type two, which are motivated from the recent work of D. Kim and T. Kim. We investigate some symmetry identities for the Changhee polynomials of type two which are derived from the properties of symmetry for the fermionic p-adic integral on Z p .