Changhee Polynomials Research Articles

Several Symmetric Identities of the Generalized Degenerate Fubini Polynomials by the Fermionic p-Adic Integral on Zp

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of many families of special polynomials and numbers, such as Bernoulli, Fubini, Bell, and Changhee polynomials and numbers. One of the main applications of these integrals is to obtain symmetric identities for the special polynomials. In this study, we focus on a novel extension of the degenerate Fubini polynomials and on obtaining some symmetric identities for them. First, we introduce the two-variable degenerate w-torsion Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. By this representation, we derive some new symmetric identities for these polynomials, using some special p-adic integral techniques. Lastly, by using some series manipulation techniques, we obtain more identities of symmetry for the two variable degenerate w-torsion Fubini polynomials.

Fractional order system of integro-differential equations: theoretical and numerical simulations using Appell-type Changhee polynomials

We examine the behavior of the solutions for a fractional-order integro-differential equation (FIDE) system using an efficient simulation presented in this paper. The FIDEs are treated by a presumably novel approximation method based on Changhee polynomials of Appell type (ACPs). In this work, we use the method of spectral collocation (SCM), which is based on the advantages of ACPs. Using this technique on the given model, it generates an algebraic equation system. Through the evaluation of the residual error function (REF), we verify the efficiency of the approach that has been offered. To verify the effectiveness and originality of the suggested algorithm, the results are compared with the precise solutions. Our results show that the method employed provides a straightforward and efficient tool to simulate the solution for such models. The suggested method's primary benefits are that it only requires a few easy steps and doesn't generate secular terms.

Theoretical treatment and implementation of the SCM included Appell-Changhee polynomials for the fractional delayed carbon absorption-emission model

Some researchers have started to perform in-depth studies on carbon dioxide emissions phenomena because global warming has caused immense damage. In this work, as a generalization of these studies, we will investigate and describe the fractional delayed Carbon absorption-emission model which consists of two fractional (Caputo sense) differential equations. From this point, we study the behavior of the solution for this model by using an approximate technique based on Appell-type Changhee polynomials (ACPs). We present an approximation of the fractional-order derivative by using ACPs. We implement the spectral collocation method (SCM) and use the properties of the ACPs to convert the provided model into a set of algebraic equations. A special focus is placed on the examination of the existence and stability of the equilibrium point for the model according to the characteristic roots with the help of the Hopf bifurcation technique. We verify the effectiveness and efficiency of the presented numerical scheme by comparing its results to those of the fourth-order Runge–Kutta method (RKM4). The results confirm that the current method is a straightforward and efficient tool for simulating the solutions to such models.

A New Family of Appell-Type Changhee Polynomials with Geometric Applications

Recently, Appell-type polynomials have been investigated and applied in several ways. In this paper, we consider a new extension of Appell-type Changhee polynomials. We introduce two-variable generalized Appell-type λ-Changhee polynomials (2VGATλCHP). The generating function, series representations, and summation identities related to these polynomials are explored. Further, certain symmetry identities involving two-variable generalized Appell-type λ-Changhee polynomials are established. Finally, Mathematica was used to examine the zero distributions of two-variable truncated-exponential Appell-type Changhee polynomials.

Non-Newtonian nanofluid flow across an exponentially stretching sheet with viscous dissipation: numerical study using an SCM based on Appell–Changhee polynomials

The objective of this article is to investigate how the properties of a non-Newtonian Williamson nanofluid flow, which occurs due to an exponential stretching sheet placed in a porous medium, are influenced by heat generation, viscous dissipation, and magnetic field. This study focuses on analyzing the heat transfer process by considering the impact of temperature on the thermal conductivity and viscosity of Williamson nanofluids. Additionally, the research significantly contributes by investigating the flow characteristics of these nanofluids when influenced by slip velocity. Using the spectral collocation method (SCM), the equations that describe the current problem are transformed into a collection of ordinary differential equations and then solved. The SCM proposed here basically depends on the properties of the Appell-type Changhee polynomials (ACPs). First, with the aid of ACPs, we give an approximate formula of the derivatives for the approximated functions. Through this procedure, the provided model is transformed into a nonlinear set of algebraic equations. Physical factors of interest, such as skin friction, the Nusselt number, and the Sherwood number, are explained using tabular expressions. Data are displayed as graphs for the nanofluid’s velocity, temperature, and concentration. The primary findings showed that increasing the Williamson, magnetic, thermal conductivity, and Brownian parameters significantly improves the thermal field. Finally, testing the suggested method with specific cases from some past literature-based publications reveal a good degree of agreement.

On Degenerate Poly-Daehee Polynomials Arising from Lambda-Umbral Calculus

In this article, we derived various identities between the degenerate poly-Daehee polynomials and some special polynomials by using λ -umbral calculus by finding the coefficients when expressing degenerate poly-Daehee polynomials as a linear combination of degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Bernoulli polynomials of the second kind, degenerate Daehee polynomials, Changhee polynomials, degenerate Bell polynomials, and degenerate Lah-Bell polynomials.

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.

Numerical investigation for the fractional model of pollution for a system of lakes using the SCM based on the Appell type Changhee polynomials

<abstract><p>This article proposed a useful simulation to investigate the Liouville-Caputo fractional order pollution model's solution behavior for a network of three lakes connected by channels. A supposedly new approximation technique using the Appell type Changhee polynomials (ACPs) was used to treat the periodic and linear input models. This work employs the spectral collocation method based on the properties of the ACPs. The given technique creates a system of algebraic equations from the studied model. We verified the efficiency of the suggested technique by computing the residual error function. We compared the results to those obtained by the fourth-order Runge-Kutta method (RK4). Our findings confirmed that the technique used provides a straightforward and efficient tool to solve such problems. The key benefit of the suggested method is that it only requires a few easy steps, doesn't produce secular terms and doesn't rely on a perturbation parameter.</p></abstract>

A Note on Higher Order Degenerate Changhee–Genocchi Numbers and Polynomials of the Second Kind

In this paper, we consider the higher order degenerate Changhee–Genocchi polynomials of the second kind by using generating functions and the Riordan matrix methods. At the same time, we give some properties of the higher order degenerate Changhee–Genocchi polynomials of the second kind. In addition, we establish some new equalities involving the higher order degenerate Changhee–Genocchi polynomials of the second kind, the generalized Bell polynomials, higher order Changhee polynomials, the higher order degenerate Daehee polynomials of the second kind, Lah numbers and Stirling numbers, etc.

Identities on Changhee Polynomials Arising from λ‐Sheffer Sequences

In this paper, authors found a new and interesting identity between Changhee polynomials and some degenerate polynomials such as degenerate Bernoulli polynomials of the first and second kind, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate Bell polynomials, degenerate Lah–Bell polynomials, and degenerate Frobenius–Euler polynomials and Mittag–Leffer polynomials by using λ‐Sheffer sequences and λ‐differential operators to find the coefficient polynomial when expressing the n‐th Changhee polynomials as a linear combination of those degenerate polynomials. In addition, authors derive the inversion formulas of these identities.

Identities of Degenerate Poly‐Changhee Polynomials Arising from λ‐Sheffer Sequences

In the 1970s, Gian‐Carlo Rota constructed the umbral calculus for investigating the properties of special functions, and by Kim‐Kim, umbral calculus is generalized called λ‐umbral calculus. In this paper, we find some important relationships between degenerate Changhee polynomials and some important special polynomials by expressing the Changhee polynomial as a linear combination of some special polynomials. In addition, we derive some interesting identities related to degenerate poly‐Changhee polynomials and some important special functions by using λ‐umbral calculus.

Appell Type Changhee Polynomials Operational Matrix of Fractional Derivatives and its Applications

Appell Type Changhee Polynomials Operational Matrix of Fractional Derivatives and its Applications

Some Identities Involving the High-Order Cauchy Polynomials

In this paper, we consider the Cauchy numbers and polynomials of order k and give some relation between Cauchy polynomials of order k and special polynomials by using generating functions and the Riordan matrix methods. In addition, we establish some new equalities and relations involving high-order Cauchy numbers and polynomials, high-order Daehee numbers and polynomials, the generalized Bell polynomials, the Bernoulli numbers and polynomials, high-order Changhee polynomials, high-order Changhee-Genocchi polynomials, the combinatorial numbers, Lah numbers and Stirling numbers, etc.

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.

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.

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.

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 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.

A note on degenerate Genocchi and poly-Genocchi numbers and polynomials

Recently, Dolgy–Jang introduced the poly-Genocchi polynomials and numbers arising from the modified polyexponential function. In this paper, we study the degenerate poly-Genocchi polynomials and numbers constructed from the modified degenerate polyexponential function. We derive explicit expressions for those polynomials and numbers. Also, we obtain identities involving those polynomials and numbers and some other special numbers and polynomials. In addition, we investigate the higher-order degenerate Genocchi polynomials and find identities involving those polynomials and the higher-order Changhee polynomials.