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23 pages, 539 KB

Open AccessArticle

On Convoluted Forms of Multivariate Legendre-Hermite Polynomials with Algebraic Matrix Based Approach

by Mumtaz Riyasat, Amal S. Alali, Shahid Ahmad Wani and Subuhi Khan

Mathematics 2024, 12(17), 2662; https://doi.org/10.3390/math12172662 - 27 Aug 2024

Viewed by 1612

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations [...] Read more.

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations comprising Hurwitz-Lerch zeta functions and

λ

-Stirling numbers of the second kind, recurrence relations, and summation formulae. The symmetry identities for these polynomials are established by connecting generalized integer power sums, double power sums and Hurwitz-Lerch zeta functions. In the end, these polynomials are also characterized Svia an algebraic matrix based approach. Full article

(This article belongs to the Section E: Applied Mathematics)

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21 pages, 826 KB

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

Cited by 7 | Viewed by 1348

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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15 pages, 737 KB

Open AccessArticle

Some Properties of Generalized Apostol-Type Frobenius–Euler–Fibonacci Polynomials

by Maryam Salem Alatawi, Waseem Ahmad Khan, Can Kızılateş and Cheon Seoung Ryoo

Mathematics 2024, 12(6), 800; https://doi.org/10.3390/math12060800 - 8 Mar 2024

Cited by 14 | Viewed by 1870

Abstract

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. [...] Read more.

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. Subsequently, we present summation formulas, Stirling–Fibonacci numbers of the second kind, and relationships for these polynomials and numbers. Finally, we define the new family of the generalized Apostol-type Frobenius–Euler–Fibonacci matrix and obtain some factorizations of this newly established matrix. Using Mathematica, the computational formulae and graphical representation for the mentioned polynomials are obtained. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

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20 pages, 1302 KB

Open AccessArticle

Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling

by Noor Alam, Waseem Ahmad Khan, Can Kızılateş, Sofian Obeidat, Cheon Seoung Ryoo and Nabawia Shaban Diab

Symmetry 2023, 15(7), 1358; https://doi.org/10.3390/sym15071358 - 4 Jul 2023

Cited by 16 | Viewed by 2125

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 study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving [...] 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 study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius–Euler polynomials of order

α

. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order

α

. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. Full article

(This article belongs to the Special Issue Special Functions and Orthogonal Polynomials: Symmetry and Applications)

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26 pages, 997 KB

Open AccessArticle

A Note on Bell-Based Apostol-Type Frobenius-Euler Polynomials of Complex Variable with Its Certain Applications

by Noor Alam, Waseem Ahmad Khan and Cheon Seoung Ryoo

Mathematics 2022, 10(12), 2109; https://doi.org/10.3390/math10122109 - 17 Jun 2022

Cited by 22 | Viewed by 2329

Abstract

In this paper, we introduce new class of Bell-based Apostol-type Frobenius–Euler polynomials and investigate some properties of these polynomials. We derive some explicit and implicit summation formulas and their symmetric identities by using different analytical means and applying generating functions of generalized Apostol-type [...] Read more.

In this paper, we introduce new class of Bell-based Apostol-type Frobenius–Euler polynomials and investigate some properties of these polynomials. We derive some explicit and implicit summation formulas and their symmetric identities by using different analytical means and applying generating functions of generalized Apostol-type Frobenius-Euler polynomials and Bell-based Apostol-type Frobenius-Euler polynomials. In particular, parametric kinds of the Bell-based Apostol-type Frobenius-Euler polynomials are introduced and some of their algebraic and analytical properties are established. In addition, illustrative examples of these families of polynomials are shown, focusing on their numerical values and piloting some beautiful computer-aided graphs of them. Full article

(This article belongs to the Special Issue Advances on Complex Analysis)

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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 7 | Viewed by 2610

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)

19 pages, 304 KB

Open AccessArticle

A New q-Hypergeometric Symbolic Calculus in the Spirit of Horn, Borngässer, Debiard and Gaveau

by Thomas Ernst

Axioms 2022, 11(2), 64; https://doi.org/10.3390/axioms11020064 - 4 Feb 2022

Cited by 4 | Viewed by 2914

Abstract

The purpose of this article is to introduce a new complete multiple q-hypergeometric symbolic calculus, which leads to q-Euler integrals and a very similar canonical system of q-difference equations for multiple q-hypergeometric functions. q-analogues of recurrence formulas in [...] Read more.

The purpose of this article is to introduce a new complete multiple q-hypergeometric symbolic calculus, which leads to q-Euler integrals and a very similar canonical system of q-difference equations for multiple q-hypergeometric functions. q-analogues of recurrence formulas in Horns paper and Borngässers thesis lead to a more exact way to find these Frobenius solutions. To find the right formulas, the parameters in q-shifted factorials can be changed to negative integers, which give no extra q-factors. In proving these q-formulas, in the limit

q \to 1

we obtain versions of the paper by Debiard and Gaveau for the solution of differential or q-difference equations. The paper is also a correction of some of the statements in the paper by Debiard and Gaveau, e.g., the Euler integrals and other solutions to differential equations for Appell functions, also without references to page numbers in the standard work of Appell and Kampé de Fériet. Sometimes the q-binomial theorem is used to simplify q-integral formulas. By the Horn method, we find another solution to the Appell

Φ_{1}

function partial differential equation, which was not mentioned in the thesis by Le Vavasseur 1893. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)

22 pages, 5257 KB

Open AccessArticle

q-Generalized Tangent Based Hybrid Polynomials

by Ghazala Yasmin, Hibah Islahi and Junesang Choi

Symmetry 2021, 13(5), 791; https://doi.org/10.3390/sym13050791 - 3 May 2021

Cited by 5 | Viewed by 2197

Abstract

In this paper, we incorporate two known polynomials to introduce so-called 2-variable q-generalized tangent based Apostol type Frobenius–Euler polynomials. Next we present a number of properties and formulas for these polynomials such as explicit expressions, series representations, summation formulas, addition formula, q [...] Read more.

In this paper, we incorporate two known polynomials to introduce so-called 2-variable q-generalized tangent based Apostol type Frobenius–Euler polynomials. Next we present a number of properties and formulas for these polynomials such as explicit expressions, series representations, summation formulas, addition formula, q-derivative and q-integral formulas, together with numerous particular cases of the new polynomials and their associated formulas demonstrated in two tables. Further, by using computer-aided programs (for example, Mathematica or Matlab), we draw graphs of some particular cases of the new polynomials, mainly, in order to observe in several angles how zeros of these polynomials are distributed and located. Lastly we provide numerous observations and questions which naturally arise amid the present investigation. Full article

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

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16 pages, 319 KB

Open AccessArticle

Some Identities of the Degenerate Higher Order Derangement Polynomials and Numbers

by Hye Kyung Kim

Symmetry 2021, 13(2), 176; https://doi.org/10.3390/sym13020176 - 22 Jan 2021

Cited by 6 | Viewed by 2275

Abstract

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the

λ

-Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials and numbers, and investigated some properties of those polynomials [...] Read more.

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the

λ

-Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials and numbers, and investigated some properties of those polynomials without using degenerate umbral calculus. In this paper, the y the degenerate derangement polynomials of order s (

s \in N

) and give a combinatorial meaning about higher order derangement numbers. In addition, the author gives some interesting identities related to the degenerate derangement polynomials of order s and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derive the inversion formulas of these identities. 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)

12 pages, 240 KB

Open AccessArticle

Generalized Tepper’s Identity and Its Application

by Dmitry Kruchinin, Vladimir Kruchinin and Yilmaz Simsek

Mathematics 2020, 8(2), 243; https://doi.org/10.3390/math8020243 - 14 Feb 2020

Cited by 5 | Viewed by 3233

Abstract

The aim of this paper is to study the Tepper identity, which is very important in number theory and combinatorial analysis. Using generating functions and compositions of generating functions, we derive many identities and relations associated with the Bernoulli numbers and polynomials, the [...] Read more.

The aim of this paper is to study the Tepper identity, which is very important in number theory and combinatorial analysis. Using generating functions and compositions of generating functions, we derive many identities and relations associated with the Bernoulli numbers and polynomials, the Euler numbers and polynomials, and the Stirling numbers. Moreover, we give applications related to the Tepper identity and these numbers and polynomials. Full article

(This article belongs to the Special Issue Special Polynomials)

13 pages, 497 KB

Open AccessArticle

Identities Associated with Generalized Stirling Type Numbers and Eulerian Type Polynomials

by Yilmaz Simsek

Math. Comput. Appl. 2013, 18(3), 251-263; https://doi.org/10.3390/mca18030251 - 1 Dec 2013

Cited by 31 | Viewed by 1979

Abstract

By using the generating functions for the generalized Stirling type numbers, Eulerian type polynomials and numbers of higher order, we derive various functional equations and differential equations. By using these equation, we derive some relations and identities related to these numbers and polynomials. [...] Read more.

By using the generating functions for the generalized Stirling type numbers, Eulerian type polynomials and numbers of higher order, we derive various functional equations and differential equations. By using these equation, we derive some relations and identities related to these numbers and polynomials. Furthermore, by applying p - adic Volkenborn integral to these polynomials, we also derive some new identities for the generalized λ-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. Full article

9 pages, 158 KB

Open AccessArticle

Generating Functions for q-Apostol Type Frobenius–Euler Numbers and Polynomials

by Yilmaz Simsek

Axioms 2012, 1(3), 395-403; https://doi.org/10.3390/axioms1030395 - 7 Dec 2012

Cited by 46 | Viewed by 7176

Abstract

The aim of this paper is to construct generating functions, related to nonnegative real parameters, for q-Eulerian type polynomials and numbers (or q-Apostol type Frobenius–Euler polynomials and numbers). We derive some identities for these polynomials and numbers based on the generating functions and [...] Read more.

The aim of this paper is to construct generating functions, related to nonnegative real parameters, for q-Eulerian type polynomials and numbers (or q-Apostol type Frobenius–Euler polynomials and numbers). We derive some identities for these polynomials and numbers based on the generating functions and functional equations. We also give multiplication formula for the generalized Apostol type Frobenius–Euler polynomials. Full article

(This article belongs to the Special Issue q-Series and Related Topics in Special Functions and Analytic Number Theory)

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