Changhee Numbers Research Articles

Symmetric Identities Involving the Extended Degenerate Central Fubini Polynomials Arising from the Fermionic p-Adic Integral on ℤp

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers and polynomials, such as central Fubini, Bernoulli, central Bell, and Changhee numbers and polynomials. One of the key applications of these integrals is for obtaining the symmetric identities of certain special polynomials. In this study, we focus on a novel generalization of degenerate central Fubini polynomials. First, we introduce two variable degenerate w-torsion central Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. Through this representation, we investigate several symmetric identities for these polynomials using special p-adic integral techniques. Also, using series manipulation methods, we obtain an identity of symmetry for the two variable degenerate w-torsion central Fubini polynomials. Finally, we provide a representation of the degenerate differential operator on the two variable degenerate w-torsion central Fubini polynomials related to the degenerate central factorial polynomials of the second kind.

Formulas for p-adic q-integrals including falling-rising factorials, combinatorial sums and special numbers

The main purpose of this paper is to provide a novel approach to deriving formulas for the p-adic q-Volkenborn integral including the Volkenborn integral and p-adic fermionic integral. By applying integral equations and these integral formulas to the falling factorials, the rising factorials and binomial coefficients, we derive some various identities, formulas and relations related to several combinatorial sums, well-known special numbers such as the Bernoulli and Euler numbers, the harmonic numbers, the Stirling numbers, the Lah numbers, the Harmonic numbers, the Fubini numbers, the Daehee numbers and the Changhee numbers. Applying these identities and formulas, we give some new combinatorial sums. Finally, by using integral equations, we derive generating functions for new families of special numbers and polynomials. By using generating functions, we give relations between the Lah numbers, the Bernoulli numbers, the Euler numbers and the Laguerre polynomials. We also give further comments and remarks on these functions, numbers and integral formulas related to q-type operators potentially used to solve problems in the areas such as physics, quantum mechanics, quantum systems and the others. In addition, we provide some tables containing some of the p-adic integral formulas obtained in this paper.

On (p,q)–Fibonacci and (p,q)–Lucas Polynomials Associated with Changhee Numbers and Their Properties

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties have been studied in the literature with the help of generating functions and their functional equations. In this paper, using the (p,q)–Fibonacci polynomials, (p,q)–Lucas polynomials, and Changhee numbers, we define the (p,q)–Fibonacci–Changhee polynomials and (p,q)–Lucas–Changhee polynomials, respectively. We obtain some important identities and relations of these newly established polynomials by using their generating functions and functional equations. Then, we generalize the (p,q)–Fibonacci–Changhee polynomials and the (p,q)–Lucas–Changhee polynomials called generalized (p,q)–Fibonacci–Lucas–Changhee polynomials. We derive a determinantal representation for the generalized (p,q)–Fibonacci–Lucas–Changhee polynomials in terms of the special Hessenberg determinant. Finally, we give a new recurrent relation of the (p,q)–Fibonacci–Lucas–Changhee polynomials.

Some Identities with Special Numbers

In this paper, we derive new identities which are related to some special numbers and generalized harmonic numbers H_n (α) by using the argument of the generating function given in [3] and comparing the coefficients of the generating functions. Also considering q -numbers involving q -Changhee numbers Chnq and q-Daehee numbers Dnq, some sums are given. For example, for any positive integer n and any positive real number q > 1, whenα= q/(q-1), we have the relationship between generalized harmonic numbers and q -Daehee numbers

A STUDY ON $q$-ANALOGUE OF DEGENERATE $\frac{1}{2}$-CHANGHEE NUMBERS AND POLYNOMIALS

The aim of the paper is to introduce $q$-analogue of degenerate $\frac{1}{2}$-Changhee numbers $Ch_{n,q,\lambda,\frac{1}{2}}$ with the help of a $p$-adic $q$-integral on $\mathbb Z_p$ and derive explicit expressions and some identities for those numbers. In more detail, we deduce explicit expressions of $Ch_{n,q,\lambda,\frac{1}{2}}$, as a rational function in terms of Euler number and Stirling numbers of the first kind, as a fermionic $p$-adic $q$-integral on $\mathbb Z_p$.

New Generalization Families of Higher Order Changhee Numbers and Polynomials

In this paper, we introduce new generalizations of higher-order Changhee of the first and second kind. Moreover, we derive some new results for these numbers and polynomials. Furthermore, some interesting special cases of the generalized higher-order Changhee numbers and polynomials are deduced.

New Families of Special Polynomial Identities Based upon Combinatorial Sums Related to p-Adic Integrals

The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive various interesting combinatorial sums and identities including new families of numbers and polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, the Changhee numbers, and other numbers and polynomials. Moreover, we present some revealing remarks and comments on the results of this paper.

Properties and Graphical Representations of the 2-Variable Form of the Simsek Polynomials

This article is written with an objective to introduce the 2-variable Simsek polynomials Yn(x,y;λ,δ) and to investigate some properties of these polynomials. The 2-variable forms of Changhee and Daehee polynomials are also considered. Several important recurrence relations involving the Cauchy, Changhee, Daehee and Simsek numbers are derived. The hypergeometric function representation for an integral involving these polynomials is obtained. The graphical representations of the 2-variable Simsek polynomials are considered for suitable values of the index n and parameters λ and δ. The article is concluded with the derivation of non-linear differential equation and related identity for the Simsek numbers.

Derivation of computational formulas for Changhee polynomials and their functional and differential equations

The goal of this paper is to demonstrate many explicit computational formulas and relations involving the Changhee polynomials and numbers and their differential equations with the help of functional equations and partial derivative equations for generating functions of these polynomials and numbers. These formulas also include the Euler polynomials, the Stirling numbers, the Bernoulli numbers and polynomials of the second kind, the Changhee polynomials of higher order, and the Daehee polynomials of higher order, which are among the well known polynomial families. By using PDEs of these generating functions, not only some recurrence relations for derivative formulas of the Changhee polynomials of higher order, but also two open problems for partial derivative equations for generating functions are given. Moreover, by using functional equations of the generating functions, two inequalities including combinatorial sums, the Changhee numbers of negative order, and the Stirling numbers of the second kind are provided. Finally, further remarks and observations for the results of this paper are given.

Some New Families of Special Polynomials and Numbers Associated with Finite Operators

The aim of this study was to define a new operator. This operator unify and modify many known operators, some of which were introduced by the author. Many properties of this operator are given. Using this operator, two new classes of special polynomials and numbers are defined. Many identities and relationships are derived, including these new numbers and polynomials, combinatorial sums, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, and the Changhee numbers. By applying the derivative operator to these new polynomials, derivative formulas are found. Integral representations, including the Volkenborn integral, the fermionic p-adic integral, and the Riemann integral, are given for these new polynomials.

Some new identities and formulas for higher-order combinatorial-type numbers and polynomials

The main purpose of this paper is to provide various identities and formulas for higherorder combinatorial-type numbers and polynomials with the help of generating functions and their both functional equations and derivative formulas. The results of this paper comprise some special numbers and polynomials such as the Stirling numbers of the first kind, the Cauchy numbers, the Changhee numbers, the Simsek numbers, the Peters poynomials, the Boole polynomials, the Simsek polynomials. Finally, remarks and observations on our results are given.

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.

Identities, inequalities for Boole-type polynomials: approach to generating functions and infinite series

The main purpose and motivation of this work is to investigate and provide some new identities, inequalities and relations for combinatorial numbers and polynomials, and for Peters type polynomials with the help of their generating functions. The results of this paper involve some special numbers and polynomials such as Stirling numbers, the Apostol–Euler numbers and polynomials, Peters polynomials, Boole polynomials, Changhee numbers and the other well-known combinatorial numbers and polynomials. Finally, in the light of Boole’s inequality (Bonferroni’s inequalities) and bounds of the Stirling numbers of the second kind, some inequalities for a combinatorial finite sum are derived. We mention an open problem including bounds for our numbers. Some remarks and observations are presented.

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.

On the degenerate (h,q)-Changhee numbers and polynomials

The Changhee numbers and polynomials are introduced by Kim, Kim and Seo (Adv. Stud. Theor. Phys. 7(20):993–1003, 2013), and the generalizations of those polynomials are characterized. In this paper, we investigate a new q-analog of the higher order degenerate Changhee polynomials and numbers. We derive some new interesting identities related to the degenerate (h,q)-Changhee polynomials and numbers.

Differential equations associated with degenerate Changhee numbers of the second kind

The degenerate Changhee numbers of the second kind are a degenerate version of Changhee numbers. Here we derive a family of nonlinear differential equations having the generating function of the degenerate Changhee numbers of the second kind as a solution. As a result, we obtain one identity involving the degenerate Changhee and higher-order degenerate Changhee numbers of the second kind.

Application of Probabilistic Method on Daehee Sequences

In this paper, we investigate some combinatorial sequences based on Daehee and Changhee numbers and polynomials, then derive their moment representations in use of probabilistic method. We also provide identities related to Daehee numbers, derangement numbers, Cauchy numbers of the second kind, and Stirling numbers of the first kind.

Some Identities Involving the Higher-Order Changhee Numbers and Polynomials

In this paper, by the classical method of Riordan arrays, establish several general involving higher-order Changhee numbers and polynomials, which are related to special polynomials and numbers. From those numbers, we derive some interesting and new identities.

Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials

The main motivation of this paper is to investigate some derivative properties of the generating functions for the numbers Yn(λ) and the polynomials Yn(x;λ), which were recently introduced by Simsek [30]. We give functional equations and differential equations (PDEs) of these generating functions. By using these functional and differential equations, we derive not only recurrence relations, but also several other identities and relations for these numbers and polynomials. Our identities include the Apostol–Bernoulli numbers, the Apostol–Euler numbers, the Stirling numbers of the first kind, the Cauchy numbers and the Hurwitz–Lerch zeta functions. Moreover, we give hypergeometric function representation for an integral involving these numbers and polynomials. Finally, we give infinite series representations of the numbers Yn(λ), the Changhee numbers, the Daehee numbers, the Lucas numbers and the Humbert polynomials.

Analysis of the p-adic q-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications

By applying the p-adic q-Volkenborn Integrals including the bosonic and the fermionic p-adic integrals on p-adic integers, we define generating functions, attached to the Dirichlet character, for the generalized Apostol-Bernoulli numbers and polynomials, the generalized Apostol-Euler numbers and polynomials, generalized Apostol-Daehee numbers and polynomials, and also generalized Apostol-Changhee numbers and polynomials. We investigate some properties of these numbers and polynomials with their generating functions. By using these generating functions and their functional equation, we give some identities and relations including the generalized Apostol-Daehee and Apostol-Changhee numbers and polynomials, the Stirling numbers, the Bernoulli numbers of the second kind, Frobenious-Euler polynomials, the generalized Bernoulli numbers and the generalized Euler numbers and the Frobenious-Euler polynomials. By using the bosonic and the fermionic p-adic integrals, we derive integral represantations for the generalized Apostol-type Daehee numbers and the generalized Apostol-type Changhee numbers.