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24 pages, 1435 KB

Open AccessArticle

Fractional Extension of Gould–Hopper–Bell Polynomials Related to Apostol-Type Polynomials and Their Properties

by Rabeb Sidaoui, Abdulghani Muhyi, Khaled Aldwoah, Khidir Shaib Mohamed, Alawia Adam, Manal Y. A. Juma and Amer Alsulami

Fractal Fract. 2026, 10(4), 244; https://doi.org/10.3390/fractalfract10040244 - 7 Apr 2026

Viewed by 359

Abstract

This study uses a fractional operator technique to analyze a novel class of special polynomials. These polynomials are designated as fractional Gould–Hopper–Bell–Apostol-type polynomials. We first define the operational expression of the Apostol-type Gould–Hopper–Bell polynomials and then use a suitable fractional operator to generate [...] Read more.

This study uses a fractional operator technique to analyze a novel class of special polynomials. These polynomials are designated as fractional Gould–Hopper–Bell–Apostol-type polynomials. We first define the operational expression of the Apostol-type Gould–Hopper–Bell polynomials and then use a suitable fractional operator to generate a new fractional version of these polynomials. The accompanying generating function, series definition, and summation formulas are also derived. Furthermore, certain symmetry identities and monomiality results are investigated. The study also identifies specific members of this fractional family, such as fractional Gould–Hopper–Bell–Apostol–Bernoulli polynomials, fractional Gould–Hopper–Bell–Apostol–Euler polynomials, and fractional Gould–Hopper–Bell–Apostol–Genocchi polynomials, and finds similar results for each. The study makes use of Mathematica to display computational results, zero distributions, and graphical demonstrations for a specific case of the established class. Full article

(This article belongs to the Section General Mathematics, Analysis)

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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 4 | Viewed by 1262

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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13 pages, 285 KB

Open AccessArticle

On Apostol-Type Hermite Degenerated Polynomials

by Clemente Cesarano, William Ramírez, Stiven Díaz, Adnan Shamaoon and Waseem Ahmad Khan

Mathematics 2023, 11(8), 1914; https://doi.org/10.3390/math11081914 - 18 Apr 2023

Cited by 17 | Viewed by 1835

Abstract

This article presents a generalization of new classes of degenerated Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi Hermite polynomials of level m. We establish some algebraic and differential properties for generalizations of new classes of degenerated Apostol–Bernoulli polynomials. These results are shown using generating function [...] Read more.

This article presents a generalization of new classes of degenerated Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi Hermite polynomials of level m. We establish some algebraic and differential properties for generalizations of new classes of degenerated Apostol–Bernoulli polynomials. These results are shown using generating function methods for Apostol–Euler and Apostol–Genocchi Hermite polynomials of level m. Full article

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

24 pages, 1495 KB

Open AccessArticle

Asymptotic Approximations of Higher-Order Apostol–Frobenius–Genocchi Polynomials with Enlarged Region of Validity

by Cristina Corcino, Wilson D. Castañeda and Roberto Corcino

Symmetry 2023, 15(4), 876; https://doi.org/10.3390/sym15040876 - 6 Apr 2023

Cited by 1 | Viewed by 2407

Abstract

In this paper, the uniform approximations of the Apostol–Frobenius–Genocchi polynomials of order α in terms of the hyperbolic functions are derived through the saddle-point method. Moreover, motivated by the works of Corcino et al., an approximation with enlarged region of validity for these [...] Read more.

In this paper, the uniform approximations of the Apostol–Frobenius–Genocchi polynomials of order α in terms of the hyperbolic functions are derived through the saddle-point method. Moreover, motivated by the works of Corcino et al., an approximation with enlarged region of validity for these polynomials is also obtained. It is found out that the methods are also applicable for the case of the higher order generalized Apostol-type Frobenius–Genocchi polynomials and Apostol–Frobenius-type poly-Genocchi polynomials with parameters a, b, and c. These methods demonstrate the techniques of computing the symmetries of the defining equation of these polynomials. Graphs are illustrated to show the accuracy of the exact values and corresponding approximations of these polynomials with respect to some specific values of its parameters. Full article

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

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10 pages, 369 KB

Open AccessEditor’s ChoiceArticle

New Classes of Degenerate Unified Polynomials

by Daniel Bedoya, Clemente Cesarano, Stiven Díaz and William Ramírez

Axioms 2023, 12(1), 21; https://doi.org/10.3390/axioms12010021 - 25 Dec 2022

Cited by 16 | Viewed by 2347

Abstract

In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating [...] Read more.

In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating function methods and we illustrate our results with some examples. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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15 pages, 342 KB

Open AccessArticle

Some Generalized Properties of Poly-Daehee Numbers and Polynomials Based on Apostol–Genocchi Polynomials

by Talha Usman, Nabiullah Khan, Mohd Aman, Shrideh Al-Omari, Kamsing Nonlaopon and Junesang Choi

Mathematics 2022, 10(14), 2502; https://doi.org/10.3390/math10142502 - 18 Jul 2022

Cited by 5 | Viewed by 2241

Abstract

Numerous polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields. The purpose of this research is to establish a unified class of Apostol–Genocchi polynomials based on poly-Daehee polynomials and to explore some of their [...] Read more.

Numerous polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields. The purpose of this research is to establish a unified class of Apostol–Genocchi polynomials based on poly-Daehee polynomials and to explore some of their features and identities. We investigate these polynomials via generating functions and deduce various identities, summation formulae, differential and integral formulas, implicit summation formulae, and several characterized generating functions for new numbers and polynomials. Finally, by using an operational version of Apostol–Genocchi polynomials, we derive some results in terms of new special polynomials. Due to the generic nature of the findings described here, they are used to reduce and generate certain known or novel formulae and identities for relatively simple polynomials and numbers. Full article

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

19 pages, 363 KB

Open AccessArticle

A Family of Generalized Legendre-Based Apostol-Type Polynomials

by Talha Usman, Nabiullah Khan, Mohd Aman and Junesang Choi

Axioms 2022, 11(1), 29; https://doi.org/10.3390/axioms11010029 - 14 Jan 2022

Cited by 6 | Viewed by 3137

Abstract

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate [...] Read more.

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials. Full article

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

23 pages, 544 KB

Open AccessArticle

Certain Results for the Twice-Iterated 2D q-Appell Polynomials

by Hari M. Srivastava, Ghazala Yasmin, Abdulghani Muhyi and Serkan Araci

Symmetry 2019, 11(10), 1307; https://doi.org/10.3390/sym11101307 - 16 Oct 2019

Cited by 24 | Viewed by 3885

Abstract

In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated [...] Read more.

In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated 2D q-Appell polynomials is also derived. Further, certain twice-iterated 2D q-Appell and mixed type special q-polynomials are considered as members of this polynomial class. The determinant expressions and some other properties of these associated members are also obtained. The graphs and surface plots of some twice-iterated 2D q-Appell and mixed type 2D q-Appell polynomials are presented for different values of indices by using Matlab. Moreover, some areas of potential applications of the subject matter of, and the results derived in, this paper are indicated. Full article

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

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20 pages, 314 KB

Open AccessArticle

A Note on the Truncated-Exponential Based Apostol-Type Polynomials

by H. M. Srivastava, Serkan Araci, Waseem A. Khan and Mehmet Acikgöz

Symmetry 2019, 11(4), 538; https://doi.org/10.3390/sym11040538 - 15 Apr 2019

Cited by 28 | Viewed by 4465

Abstract

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric [...] Read more.

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

14 pages, 306 KB

Open AccessArticle

Higher-Order Convolutions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials

by Yuan He, Serkan Araci, Hari M. Srivastava and Mahmoud Abdel-Aty

Mathematics 2018, 6(12), 329; https://doi.org/10.3390/math6120329 - 14 Dec 2018

Cited by 15 | Viewed by 3497

Abstract

In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the [...] Read more.

In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas. Full article

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

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