MDPI - Publisher of Open Access Journals

19 pages, 392 KiB

Open AccessArticle

Szász–Beta Operators Linking Frobenius–Euler–Simsek-Type Polynomials

by Nadeem Rao, Mohammad Farid and Shivani Bansal

Axioms 2025, 14(6), 418; https://doi.org/10.3390/axioms14060418 - 29 May 2025

Viewed by 293

This manuscript associates with a study of Frobenius–Euler–Simsek-type Polynomials. In this research work, we construct a new sequence of Szász–Beta type operators via Frobenius–Euler–Simsek-type Polynomials to discuss approximation properties for the Lebesgue integrable functions, i.e.,

L^{p} [0, \infty)

, [...] Read more.

This manuscript associates with a study of Frobenius–Euler–Simsek-type Polynomials. In this research work, we construct a new sequence of Szász–Beta type operators via Frobenius–Euler–Simsek-type Polynomials to discuss approximation properties for the Lebesgue integrable functions, i.e.,

L^{p} [0, \infty)

,

1 \leq p < \infty

. Furthermore, estimates in view of test functions and central moments are studied. Next, rate of convergence is discussed with the aid of the Korovkin theorem and the Voronovskaja type theorem. Moreover, direct approximation results in terms of modulus of continuity of first- and second-order, Peetre’s K-functional, Lipschitz type space, and the

r^{t h}

-order Lipschitz type maximal functions are investigated. In the subsequent section, we present weighted approximation results, and statistical approximation theorems are discussed. To demonstrate the effectiveness and applicability of the proposed operators, we present several illustrative examples and visualize the results graphically. Full article

(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)

► Show Figures

Figure 1

15 pages, 649 KiB

Open AccessArticle

On the Approximations and Symmetric Properties of Frobenius–Euler–Şimşek Polynomials Connecting Szász Operators

by Nadeem Rao, Mohammad Farid and Mohd Raiz

Symmetry 2025, 17(5), 648; https://doi.org/10.3390/sym17050648 - 25 Apr 2025

Cited by 2 | Viewed by 309

Abstract

This study focuses on approximating continuous functions using Frobenius–Euler–Simsek polynomial analogues of Szász operators. Test functions and central moments are computed to study convergence uniformly, approximation order by these operators. Next, we investigate approximation order uniform convergence via Korovkin result and the modulus [...] Read more.

This study focuses on approximating continuous functions using Frobenius–Euler–Simsek polynomial analogues of Szász operators. Test functions and central moments are computed to study convergence uniformly, approximation order by these operators. Next, we investigate approximation order uniform convergence via Korovkin result and the modulus of smoothness for functions in continuous functional spaces. A Voronovskaja theorem is also explored approximating functions which belongs to the class of function having first and second order continuous derivative. Further, we discuss numerical error and graphical analysis. In the last, two dimensional operators are constructed to discuss approximation for the class of two variable continuous functions. Full article

(This article belongs to the Special Issue Symmetries and Fractional Differential Equations: Theory and Applications)

► Show Figures

Figure 1

15 pages, 507 KiB

Open AccessArticle

Truncated-Exponential-Based General-Appell Polynomials

by Zeynep Özat, Bayram Çekim, Mehmet Ali Özarslan and Francesco Aldo Costabile

Mathematics 2025, 13(8), 1266; https://doi.org/10.3390/math13081266 - 11 Apr 2025

Cited by 3 | Viewed by 363

Abstract

In this paper, a new and general form of truncated-exponential-based general-Appell polynomials is introduced using the two-variable general-Appell polynomials. For this new polynomial family, we present an explicit representation, recurrence relation, shift operators, differential equation, determinant representation, and some other properties. Finally, two [...] Read more.

In this paper, a new and general form of truncated-exponential-based general-Appell polynomials is introduced using the two-variable general-Appell polynomials. For this new polynomial family, we present an explicit representation, recurrence relation, shift operators, differential equation, determinant representation, and some other properties. Finally, two special cases of this family, truncated-exponential-based Hermite-type and truncated-exponential-based Laguerre–Frobenius Euler polynomials, are introduced and their corresponding properties are obtained. Full article

(This article belongs to the Section C: Mathematical Analysis)

► Show Figures

Figure 1

15 pages, 660 KiB

Open AccessArticle

Approximation Results: Szász–Kantorovich Operators Enhanced by Frobenius–Euler–Type Polynomials

by Nadeem Rao, Mohammad Farid and Mohd Raiz

Axioms 2025, 14(4), 252; https://doi.org/10.3390/axioms14040252 - 27 Mar 2025

Cited by 3 | Viewed by 382

Abstract

This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the [...] Read more.

This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the modulus of continuity for Lebesgue measurable and continuous functions. A Voronovskaja-type theorem is also established to approximate functions with first- and second-order continuous derivatives. Numerical and graphical analyses are presented to support these findings. Furthermore, a bivariate sequence of these operators is introduced to approximate a bivariate class of Lebesgue measurable and continuous functions in two variables. Finally, numerical and graphical representations of the error are provided to check the rapidity of convergence. Full article

(This article belongs to the Special Issue Numerical Methods and Approximation Theory)

► Show Figures

Figure 1

23 pages, 539 KiB

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 872

Abstract

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)

► Show Figures

Figure 1

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 896

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)

► Show Figures

Figure 1

15 pages, 737 KiB

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 5 | Viewed by 1272

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)

► Show Figures

Figure 1

17 pages, 321 KiB

Open AccessArticle

Properties of Multivariate Hermite Polynomials in Correlation with Frobenius–Euler Polynomials

by Mohra Zayed, Shahid Ahmad Wani and Yamilet Quintana

Mathematics 2023, 11(16), 3439; https://doi.org/10.3390/math11163439 - 8 Aug 2023

Cited by 10 | Viewed by 1486

Abstract

A comprehensive framework has been developed to apply the monomiality principle from mathematical physics to various mathematical concepts from special functions. This paper presents research on a novel family of multivariate Hermite polynomials associated with Apostol-type Frobenius–Euler polynomials. The study derives the generating [...] Read more.

A comprehensive framework has been developed to apply the monomiality principle from mathematical physics to various mathematical concepts from special functions. This paper presents research on a novel family of multivariate Hermite polynomials associated with Apostol-type Frobenius–Euler polynomials. The study derives the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified. Moreover, the research establishes series representations, summation formulae, and operational and symmetric identities, as well as recurrence relations satisfied by these polynomials. Full article

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

20 pages, 1302 KiB

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 10 | Viewed by 1479

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)

► Show Figures

Figure 1

21 pages, 996 KiB

Open AccessArticle

Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling

by Yongsheng Rao, Waseem Ahmad Khan, Serkan Araci and Cheon Seoung Ryoo

Mathematics 2023, 11(10), 2386; https://doi.org/10.3390/math11102386 - 20 May 2023

Cited by 7 | Viewed by 1487

Abstract

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the [...] Read more.

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

► Show Figures

Figure 1

18 pages, 1853 KiB

Open AccessArticle

Closed-Form Solution for the Natural Frequencies of Low-Speed Cracked Euler–Bernoulli Rotating Beams

by Belén Muñoz-Abella, Lourdes Rubio and Patricia Rubio

Mathematics 2022, 10(24), 4742; https://doi.org/10.3390/math10244742 - 14 Dec 2022

Cited by 2 | Viewed by 2064

Abstract

In this study, two closed-form solutions for determining the first two natural frequencies of the flapwise bending vibration of a cracked Euler–Bernoulli beam at low rotational speed have been developed. To solve the governing differential equations of motion, the Frobenius method of solution [...] Read more.

In this study, two closed-form solutions for determining the first two natural frequencies of the flapwise bending vibration of a cracked Euler–Bernoulli beam at low rotational speed have been developed. To solve the governing differential equations of motion, the Frobenius method of solution in power series has been used. The crack has been modeled using two undamaged parts of the beam connected by a rotational spring. From the previous results, two novel polynomial expressions have been developed to obtain the first two natural frequencies as a function of angular velocity, slenderness ratio, cube radius and crack characteristics (depth and location). These expressions have been formulated using multiple regression techniques. To the knowledge of the authors, there is no similar expressions in the literature, which calculate, in a simple way, the first two natural frequencies based on beam features and crack parameters, without the need to know or solve the differential equations of motion governing the beam. In summary, the derived natural frequency expressions provide an extremely simple, practical, and accurate instrument for studying the dynamic behavior of rotating cracked Euler–Bernoulli beams at low angular speed, especially useful, in the future, to establish small-scale wind turbines’ maintenance planes. Full article

(This article belongs to the Section E2: Control Theory and Mechanics)

► Show Figures

Figure 1

13 pages, 273 KiB

Open AccessArticle

Fourier Series Expansion and Integral Representation of Apostol-Type Frobenius–Euler Polynomials of Complex Parameters and Order α

by Cristina Corcino, Roberto Corcino and Jeremar Casquejo

Symmetry 2022, 14(9), 1860; https://doi.org/10.3390/sym14091860 - 6 Sep 2022

Cited by 3 | Viewed by 1956

Abstract

In this paper, the Fourier series expansions of Apostol-type Frobenius–Euler polynomials of complex parameters and order

α

are derived, and consequently integral representations of these polynomials are established. This paper provides some techniques in computing the symmetries of the defining equation of Apostol-type [...] Read more.

In this paper, the Fourier series expansions of Apostol-type Frobenius–Euler polynomials of complex parameters and order

α

are derived, and consequently integral representations of these polynomials are established. This paper provides some techniques in computing the symmetries of the defining equation of Apostol-type Frobenius–Euler polynomials resulting in their expansions and integral representations. Full article

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

26 pages, 997 KiB

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 18 | Viewed by 1908

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)

► Show Figures

Figure 1

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 2182

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)

22 pages, 5257 KiB

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 1800

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)

► Show Figures

Figure 1

Search Results (20)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (20)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI