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12 pages, 275 KiB

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

Viewed by 802

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)

21 pages, 826 KiB

Open AccessArticle

On Certain Properties of Parametric Kinds of Apostol-Type Frobenius–Euler–Fibonacci Polynomials

by Hao Guan, Waseem Ahmad Khan, Can Kızılateş and Cheon Seoung Ryoo

Axioms 2024, 13(6), 348; https://doi.org/10.3390/axioms13060348 - 24 May 2024

Viewed by 795

Abstract

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving [...] Read more.

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials of order

$α$ and several other polynomial sequences, such as the Apostol-type Bernoulli–Fibonacci polynomials, the Apostol-type Euler–Fibonacci polynomials, the Apostol-type Genocchi–Fibonacci polynomials, and the Stirling–Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program. Full article

(This article belongs to the Special Issue Fractional and Stochastic Differential Equations in Mathematics)

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18 pages, 1242 KiB

Open AccessArticle

Novel Properties of q-Sine-Based and q-Cosine-Based q-Fubini Polynomials

by Waseem Ahmad Khan, Maryam Salem Alatawi, Cheon Seoung Ryoo and Ugur Duran

Symmetry 2023, 15(2), 356; https://doi.org/10.3390/sym15020356 - 28 Jan 2023

Cited by 8 | Viewed by 1275

Abstract

The main purpose of this paper is to consider q-sine-based and q-cosine-Based q-Fubini polynomials and is to investigate diverse properties of these polynomials. Furthermore, multifarious correlations including q-analogues of the Genocchi, Euler and Bernoulli polynomials, and the q-Stirling [...] Read more.

The main purpose of this paper is to consider q-sine-based and q-cosine-Based q-Fubini polynomials and is to investigate diverse properties of these polynomials. Furthermore, multifarious correlations including q-analogues of the Genocchi, Euler and Bernoulli polynomials, and the q-Stirling numbers of the second kind are derived. Moreover, some approximate zeros of the q-sinebased and q-cosine-Based q-Fubini polynomials in a complex plane are examined, and lastly, these zeros are shown using figures. Full article

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

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14 pages, 282 KiB

Open AccessArticle

A Note on Degenerate Catalan-Daehee Numbers and Polynomials

by Waseem Ahmad Khan, Maryam Salem Alatawi and Ugur Duran

Symmetry 2022, 14(10), 2169; https://doi.org/10.3390/sym14102169 - 16 Oct 2022

Cited by 4 | Viewed by 1416

Abstract

In this paper, we consider the degenerate forms of the Catalan–Daehee polynomials and numbers by the Volkenborn integrals and obtain diverse explicit expressions and formulas. Moreover, we show the expressions of the degenerate Catalan–Daehee numbers in terms of

$λ$ -Daehee numbers, Stirling numbers [...] Read more.

In this paper, we consider the degenerate forms of the Catalan–Daehee polynomials and numbers by the Volkenborn integrals and obtain diverse explicit expressions and formulas. Moreover, we show the expressions of the degenerate Catalan–Daehee numbers in terms of

$λ$ -Daehee numbers, Stirling numbers of the first kind and Bernoulli polynomials, and we also obtain a relation covering the Bernoulli numbers, the degenerate Catalan–Daehee numbers and Stirling numbers of the second kind. In addition, we prove an implicit summation formula and a symmetric identity, and we derive an explicit expression for the degenerate Catalan–Daehee polynomials including the Stirling numbers of the first kind and Bernoulli polynomials. Full article

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

12 pages, 322 KiB

Open AccessArticle

Lagrange-Based Hypergeometric Bernoulli Polynomials

by Sahar Albosaily, Yamilet Quintana, Azhar Iqbal and Waseem A. Khan

Symmetry 2022, 14(6), 1125; https://doi.org/10.3390/sym14061125 - 30 May 2022

Cited by 6 | Viewed by 1821

Abstract

Special polynomials play an important role in several subjects of mathematics, engineering, and theoretical physics. Many problems arising in mathematics, engineering, and mathematical physics are framed in terms of differential equations. In this paper, we introduce the family of the Lagrange-based hypergeometric Bernoulli [...] Read more.

Special polynomials play an important role in several subjects of mathematics, engineering, and theoretical physics. Many problems arising in mathematics, engineering, and mathematical physics are framed in terms of differential equations. In this paper, we introduce the family of the Lagrange-based hypergeometric Bernoulli polynomials via the generating function method. We state some algebraic and differential properties for this family of extensions of the Lagrange-based Bernoulli polynomials, as well as a matrix-inversion formula involving these polynomials. Moreover, a generating relation involving the Stirling numbers of the second kind was derived. In fact, future investigations in this subject could be addressed for the potential applications of these polynomials in the aforementioned disciplines. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

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26 pages, 379 KiB

Open AccessArticle

Note on the Higher-Order Derivatives of the Hyperharmonic Polynomials and the r-Stirling Polynomials of the First Kind

by José L. Cereceda

Axioms 2022, 11(4), 167; https://doi.org/10.3390/axioms11040167 - 8 Apr 2022

Viewed by 2155

Abstract

In this paper, we focus on the higher-order derivatives of the hyperharmonic polynomials, which are a generalization of the ordinary harmonic numbers. We determine the hyperharmonic polynomials and their successive derivatives in terms of the r-Stirling polynomials of the first kind and [...] Read more.

In this paper, we focus on the higher-order derivatives of the hyperharmonic polynomials, which are a generalization of the ordinary harmonic numbers. We determine the hyperharmonic polynomials and their successive derivatives in terms of the r-Stirling polynomials of the first kind and show the relationship between the (exponential) complete Bell polynomials and the r-Stirling numbers of the first kind. Furthermore, we provide a new formula for obtaining the generalized Bernoulli polynomials by exploiting their link with the higher-order derivatives of the hyperharmonic polynomials. In addition, we obtain various identities involving the r-Stirling numbers of the first kind, the Bernoulli numbers and polynomials, the Stirling numbers of the first and second kind, and the harmonic numbers. Full article

(This article belongs to the Section Mathematical Analysis)

12 pages, 290 KiB

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 2095

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)

16 pages, 1881 KiB

Open AccessArticle

A Family of the r-Associated Stirling Numbers of the Second Kind and Generalized Bernoulli Polynomials

by Paolo Emilio Ricci, Rekha Srivastava and Pierpaolo Natalini

Axioms 2021, 10(3), 219; https://doi.org/10.3390/axioms10030219 - 9 Sep 2021

Cited by 5 | Viewed by 3311

Abstract

In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links [...] Read more.

In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links with available literature on this subject are also pointed out. The extension to the bivariate case is discussed. Full article

(This article belongs to the Collection Mathematical Analysis and Applications)

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23 pages, 340 KiB

Open AccessFeature PaperArticle

Bell-Based Bernoulli Polynomials with Applications

by Ugur Duran, Serkan Araci and Mehmet Acikgoz

Axioms 2021, 10(1), 29; https://doi.org/10.3390/axioms10010029 - 2 Mar 2021

Cited by 14 | Viewed by 3234

Abstract

In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order [...] Read more.

In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order

$?$ and investigate multifarious correlations and formulas including some summation formulas and derivative properties. Also, we acquire diverse implicit summation formulas and symmetric identities for Bell-based Bernoulli polynomials of order

$?$ . Moreover, we attain several interesting formulas of Bell-based Bernoulli polynomials of order

$?$ arising from umbral calculus. Full article

(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)

13 pages, 282 KiB

Open AccessArticle

Two-Variable Type 2 Poly-Fubini Polynomials

by Ghulam Muhiuddin, Waseem Ahmad Khan and Ugur Duran

Mathematics 2021, 9(3), 281; https://doi.org/10.3390/math9030281 - 31 Jan 2021

Cited by 16 | Viewed by 2374

Abstract

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first [...] Read more.

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

15 pages, 244 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Type 2 Degenerate Poly-Euler Polynomials

by Dae Sik Lee, Hye Kyung Kim and Lee-Chae Jang

Symmetry 2020, 12(6), 1011; https://doi.org/10.3390/sym12061011 - 15 Jun 2020

Cited by 12 | Viewed by 2860

Abstract

In recent years, many mathematicians have studied the degenerate versions of many special polynomials and numbers. The polyexponential functions were introduced by Hardy and rediscovered by Kim, as inverses to the polylogarithms functions. The paper is divided two parts. First, we introduce a [...] Read more.

In recent years, many mathematicians have studied the degenerate versions of many special polynomials and numbers. The polyexponential functions were introduced by Hardy and rediscovered by Kim, as inverses to the polylogarithms functions. The paper is divided two parts. First, we introduce a new type of the type 2 poly-Euler polynomials and numbers constructed from the modified polyexponential function, the so-called type 2 poly-Euler polynomials and numbers. We show various expressions and identities for these polynomials and numbers. Some of them involving the (poly) Euler polynomials and another special numbers and polynomials such as (poly) Bernoulli polynomials, the Stirling numbers of the first kind, the Stirling numbers of the second kind, etc. In final section, we introduce a new type of the type 2 degenerate poly-Euler polynomials and the numbers defined in the previous section. We give explicit expressions and identities involving those polynomials in a similar direction to the previous section. Full article

(This article belongs to the Special Issue The 33rd International Conference of The Jangjeon Mathematical Society (ICJMS2020) will be held at Hasanuddin University, Makassar, Indonesia)

16 pages, 245 KiB

Open AccessArticle

A Parametric Kind of Fubini Polynomials of a Complex Variable

by Sunil Kumar Sharma, Waseem A. Khan and Cheon Seoung Ryoo

Mathematics 2020, 8(4), 643; https://doi.org/10.3390/math8040643 - 22 Apr 2020

Cited by 5 | Viewed by 2305

Abstract

In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in [...] Read more.

In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in a systematic way. Furthermore, we consider some relationships for parametric kind of Fubini polynomials associated with Bernoulli, Euler, and Genocchi polynomials and Stirling numbers of the second kind. Full article

(This article belongs to the Special Issue Special Polynomials)

9 pages, 248 KiB

Open AccessArticle

Some Identities on Type 2 Degenerate Bernoulli Polynomials of the Second Kind

by Taekyun Kim, Lee-Chae Jang, Dae San Kim and Han Young Kim

Symmetry 2020, 12(4), 510; https://doi.org/10.3390/sym12040510 - 2 Apr 2020

Cited by 25 | Viewed by 2574

Abstract

In recent years, many mathematicians studied various degenerate versions of some special polynomials for which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study [...] Read more.

In recent years, many mathematicians studied various degenerate versions of some special polynomials for which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specifically, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of the second kind, and an expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind. Full article

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

18 pages, 247 KiB

Open AccessArticle

On Two Bivariate Kinds of Poly-Bernoulli and Poly-Genocchi Polynomials

by Cheon Seoung Ryoo and Waseem A. Khan

Mathematics 2020, 8(3), 417; https://doi.org/10.3390/math8030417 - 14 Mar 2020

Cited by 17 | Viewed by 1963

Abstract

In this paper, we introduce two bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials and study their basic properties. Finally, we consider some relationships for Stirling numbers of the second kind related to bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials. Full article

43 pages, 1041 KiB

Open AccessArticle

On Degenerate Truncated Special Polynomials

by Ugur Duran and Mehmet Acikgoz

Mathematics 2020, 8(1), 144; https://doi.org/10.3390/math8010144 - 20 Jan 2020

Cited by 13 | Viewed by 3830

Abstract

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential [...] Read more.

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential polynomials are first considered and their several properties are given. Then the degenerate truncated Stirling polynomials of the second kind are defined and their elementary properties and relations are proved. Also, the degenerate truncated forms of the bivariate Fubini and Bell polynomials and numbers are introduced and various relations and formulas for these polynomials and numbers, which cover several summation formulas, addition identities, recurrence relationships, derivative property and correlations with the degenerate truncated Stirling polynomials of the second kind, are acquired. Thereafter, the truncated degenerate Bernoulli and Euler polynomials are considered and multifarious correlations and formulas including summation formulas, derivation rules and correlations with the degenerate truncated Stirling numbers of the second are derived. In addition, regarding applications, by introducing the degenerate truncated forms of the classical Bernstein polynomials, we obtain diverse correlations and formulas. Some interesting surface plots of these polynomials in the special cases are provided. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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