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12 pages, 275 KB

Open AccessArticle

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

by Maryam Salem Alatawi, Waseem Ahmad Khan and Ugur Duran

Axioms 2024, 13(7), 421; https://doi.org/10.3390/axioms13070421 - 22 Jun 2024

Cited by 1 | Viewed by 1067

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications, 2nd Edition)

11 pages, 264 KB

Open AccessArticle

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

Z_{p}

by Maryam Salem Alatawi, Waseem Ahmad Khan and Ugur Duran

Symmetry 2024, 16(6), 686; https://doi.org/10.3390/sym16060686 - 3 Jun 2024

Cited by 1 | Viewed by 885

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

10 pages, 283 KB

Open AccessArticle

On Generalized Bivariate (p,q)-Bernoulli–Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli–Lucas Polynomials

by Hao Guan, Waseem Ahmad Khan and Can Kızılateş

Symmetry 2023, 15(4), 943; https://doi.org/10.3390/sym15040943 - 20 Apr 2023

Cited by 9 | Viewed by 1890

Abstract

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, we define the generalized

(p, q)

-Bernoulli–Fibonacci [...] Read more.

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, we define the generalized

(p, q)

-Bernoulli–Fibonacci and generalized

(p, q)

-Bernoulli–Lucas polynomials and numbers by using the

(p, q)

-Bernoulli numbers, unified

(p, q)

-Bernoulli polynomials,

h (x)

-Fibonacci polynomials, and

h (x)

-Lucas polynomials. We also introduce the generalized bivariate

(p, q)

-Bernoulli–Fibonacci and generalized bivariate

(p, q)

-Bernoulli–Lucas polynomials and numbers. Then, we derive some properties of these newly established polynomials and numbers by using their generating functions with their functional equations. Finally, we provide some families of bilinear and bilateral generating functions for the generalized bivariate

(p, q)

-Bernoulli–Fibonacci polynomials. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

18 pages, 1114 KB

Open AccessArticle

Diverse Properties and Approximate Roots for a Novel Kinds of the (p,q)-Cosine and (p,q)-Sine Geometric Polynomials

by Sunil Kumar Sharma, Waseem Ahmad Khan, Cheon-Seoung Ryoo and Ugur Duran

Mathematics 2022, 10(15), 2709; https://doi.org/10.3390/math10152709 - 31 Jul 2022

Cited by 3 | Viewed by 1750

Abstract

Utilizing

(p, q)

-numbers and

(p, q)

-concepts, in 2016, Duran et al. considered

(p, q)

-Genocchi numbers and polynomials,

(p, q)

-Bernoulli numbers and polynomials and

(p, q)

-Euler polynomials and numbers and provided multifarious formulas and [...] Read more.

Utilizing

(p, q)

-numbers and

(p, q)

-concepts, in 2016, Duran et al. considered

(p, q)

-Genocchi numbers and polynomials,

(p, q)

-Bernoulli numbers and polynomials and

(p, q)

-Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. Inspired and motivated by this consideration, many authors have introduced

(p, q)

-special polynomials and numbers and have described some of their properties and applications. In this paper, using the

(p, q)

-cosine polynomials and

(p, q)

-sine polynomials, we consider a novel kinds of

(p, q)

-extensions of geometric polynomials and acquire several properties and identities by making use of some series manipulation methods. Furthermore, we compute the

(p, q)

-integral representations and

(p, q)

-derivative operator rules for the new polynomials. Additionally, we determine the movements of the approximate zerosof the two mentioned polynomials in a complex plane, utilizing the Newton method, and we illustrate them using figures. Full article

(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)

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12 pages, 290 KB

Open AccessArticle

On (p, q)-Sine and (p, q)-Cosine Fubini Polynomials

by Waseem Ahmad Khan, Ghulam Muhiuddin, Ugur Duran and Deena Al-Kadi

Symmetry 2022, 14(3), 527; https://doi.org/10.3390/sym14030527 - 4 Mar 2022

Cited by 6 | Viewed by 2361

Abstract

In recent years,

(p, q)

-special polynomials, such as

(p, q)

-Euler,

(p, q)

-Genocchi,

(p, q)

-Bernoulli, and

(p, q)

-Frobenius-Euler, have been studied and investigated by many mathematicians, as well physicists. It is important [...] Read more.

In recent years,

(p, q)

-special polynomials, such as

(p, q)

-Euler,

(p, q)

-Genocchi,

(p, q)

-Bernoulli, and

(p, q)

-Frobenius-Euler, have been studied and investigated by many mathematicians, as well physicists. It is important that any polynomial have explicit formulas, symmetric identities, summation formulas, and relations with other polynomials. In this work, the

(p, q)

-sine and

(p, q)

-cosine Fubini polynomials are introduced and multifarious abovementioned properties for these polynomials are derived by utilizing some series manipulation methods.

(p, q)

-derivative operator rules and

(p, q)

-integral representations for the

(p, q)

-sine and

(p, q)

-cosine Fubini polynomials are also given. Moreover, several correlations related to both the

(p, q)

-Bernoulli, Euler, and Genocchi polynomials and the

(p, q)

-Stirling numbers of the second kind are developed. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

21 pages, 939 KB

Open AccessFeature PaperArticle

Structure of Approximate Roots Based on Symmetric Properties of (p, q)-Cosine and (p, q)-Sine Bernoulli Polynomials

by Cheon Seoung Ryoo and Jung Yoog Kang

Symmetry 2020, 12(6), 885; https://doi.org/10.3390/sym12060885 - 30 May 2020

Cited by 4 | Viewed by 2411

Abstract

This paper constructs and introduces

(p, q)

-cosine and

(p, q)

-sine Bernoulli polynomials using

(p, q)

-analogues of

{(x + a)}^{n}

. Based on these polynomials, we discover basic properties [...] Read more.

This paper constructs and introduces

(p, q)

-cosine and

(p, q)

-sine Bernoulli polynomials using

(p, q)

-analogues of

{(x + a)}^{n}

. Based on these polynomials, we discover basic properties and identities. Moreover, we determine special properties using

(p, q)

-trigonometric functions and verify various symmetric properties. Finally, we check the symmetric structure of the approximate roots based on symmetric polynomials. Full article

(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)

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10 pages, 255 KB

Open AccessArticle

Note on Type 2 Degenerate q-Bernoulli Polynomials

by Dae San Kim, Dmitry V. Dolgy, Jongkyum Kwon and Taekyun Kim

Symmetry 2019, 11(7), 914; https://doi.org/10.3390/sym11070914 - 13 Jul 2019

Cited by 4 | Viewed by 2430

Abstract

The purpose of this paper is to introduce and study type 2 degenerate q-Bernoulli polynomials and numbers by virtue of the bosonic p-adic q-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, [...] Read more.

The purpose of this paper is to introduce and study type 2 degenerate q-Bernoulli polynomials and numbers by virtue of the bosonic p-adic q-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, identities regarding Carlitz’s q-Bernoulli numbers, identities concerning degenerate q-Bernoulli numbers, and the representations of the fully degenerate type 2 Bernoulli numbers in terms of moments of certain random variables, created from random variables with Laplace distributions. It is expected that, as was done in the case of type 2 degenerate Bernoulli polynomials and numbers, we will be able to find some identities of symmetry for those polynomials and numbers. Full article

(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)

10 pages, 255 KB

Open AccessArticle

Symmetric Identities of Hermite-Bernoulli Polynomials and Hermite-Bernoulli Numbers Attached to a Dirichlet Character χ

by Serkan Araci, Waseem Ahmad Khan and Kottakkaran Sooppy Nisar

Symmetry 2018, 10(12), 675; https://doi.org/10.3390/sym10120675 - 29 Nov 2018

Cited by 4 | Viewed by 2838

Abstract

We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials and Hermite-Bernoulli numbers attached to a Dirichlet character

χ

and investigate certain symmetric identities involving the polynomials, by mainly using the theory of p-adic integral on

Z_{p}

. The results presented here, [...] Read more.

We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials and Hermite-Bernoulli numbers attached to a Dirichlet character

χ

and investigate certain symmetric identities involving the polynomials, by mainly using the theory of p-adic integral on

Z_{p}

. The results presented here, being very general, are shown to reduce to yield symmetric identities for many relatively simple polynomials and numbers and some corresponding known symmetric identities. Full article

(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications)

11 pages, 251 KB

Open AccessArticle

On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

by Dae San Kim, Taekyun Kim, Cheon Seoung Ryoo and Yonghong Yao

Symmetry 2018, 10(10), 451; https://doi.org/10.3390/sym10100451 - 1 Oct 2018

Cited by 2 | Viewed by 2746

Abstract

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on

Z_{p}

. We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties [...] Read more.

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on

Z_{p}

. We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on

Z_{p}

of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found. Full article

(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications)

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