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18 pages, 318 KiB

Open AccessArticle

On Analytical Extension of Generalized Hypergeometric Function ₃F₂

by Roman Dmytryshyn and Volodymyra Oleksyn

Axioms 2024, 13(11), 759; https://doi.org/10.3390/axioms13110759 - 31 Oct 2024

Viewed by 369

Abstract

The paper considers the generalized hypergeometric function

{}_{3}F_{2},

which is important in various fields of mathematics, physics, and economics. The method is used, according to which the domains of the analytical continuation of the special functions are the domains of [...] Read more.

The paper considers the generalized hypergeometric function

{}_{3}F_{2},

which is important in various fields of mathematics, physics, and economics. The method is used, according to which the domains of the analytical continuation of the special functions are the domains of convergence of their expansions into a special family of functions, namely branched continued fractions. These expansions have wide domains of convergence and better computational properties, particularly compared with series, making them effective tools for representing special functions. New domains of the analytical continuation of the generalized hypergeometric function

{}_{3}F_{2}

with real and complex parameters have been established. The paper also includes examples of the presentation and extension of some special functions. Full article

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

15 pages, 277 KiB

Open AccessArticle

Exchange Formulae for the Stieltjes–Poisson Transform over Weighted Lebesgue Spaces

by Hari M. Srivastava, Emilio Ramón Negrín and Jeetendrasingh Maan

Axioms 2024, 13(11), 748; https://doi.org/10.3390/axioms13110748 - 30 Oct 2024

Viewed by 398

Abstract

This paper aims to develop exchange formulae for the Stieltjes–Poisson transform by using Mellin-type convolutions in the context of weighted Lebesgue spaces. A key result is the introduction of bilinear and continuous Mellin-type convolutions, expanding the scope of the analysis to include the [...] Read more.

This paper aims to develop exchange formulae for the Stieltjes–Poisson transform by using Mellin-type convolutions in the context of weighted Lebesgue spaces. A key result is the introduction of bilinear and continuous Mellin-type convolutions, expanding the scope of the analysis to include the space of weighted

L^{1}

functions and the space of continuous functions vanishing at infinity. Full article

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

11 pages, 257 KiB

Open AccessArticle

Parseval–Goldstein-Type Theorems for Lebedev–Skalskaya Transforms

by Emilio Ramón Negrín, Benito Juan González and Jeetendrasingh Maan

Axioms 2024, 13(9), 630; https://doi.org/10.3390/axioms13090630 - 14 Sep 2024

Viewed by 546

Abstract

This paper investigates Parseval–Goldstein-type relationships in the framework of Lebedev–Skalskaya transforms. The research also examines the continuity properties of these transforms, along with their adjoint counterparts over weighted Lebesgue spaces. Furthermore, the behavior of Lebedev–Skalskaya transforms and their adjoint transforms in the context [...] Read more.

This paper investigates Parseval–Goldstein-type relationships in the framework of Lebedev–Skalskaya transforms. The research also examines the continuity properties of these transforms, along with their adjoint counterparts over weighted Lebesgue spaces. Furthermore, the behavior of Lebedev–Skalskaya transforms and their adjoint transforms in the context of weighted Lebesgue spaces is analyzed. This study aims to provide deeper insights into the functional properties and applications of these transforms in mathematical analysis. Full article

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

16 pages, 275 KiB

Open AccessArticle

Fourier Series Related to p-Trigonometric Functions

by Ali Hamzah Alibrahim and Saptarshi Das

Axioms 2024, 13(9), 600; https://doi.org/10.3390/axioms13090600 - 4 Sep 2024

Viewed by 621

Abstract

In this paper, we introduce the concept of generalized Fourier series, generated by the p-trigonometric functions, namely cosp and sinp, recently introduced related to the generalized complex numbers systems. The aim of this study is to represent a periodic signal as a [...] Read more.

In this paper, we introduce the concept of generalized Fourier series, generated by the p-trigonometric functions, namely cosp and sinp, recently introduced related to the generalized complex numbers systems. The aim of this study is to represent a periodic signal as a sum of p-sine and p-cosine functions. In order to achieve this, we first present the integrals of the product of the same or different family of p-trigonometric functions over the full period of these functions to understand the orthogonality properties. Next, we use these integrals to derive the coefficients of the generalized p-Fourier series along with a few examples. The generalized Fourier series can be used to expand an arbitrary forcing function in the solution of a non-homogeneous linear ordinary differential equation (ODE) with constant coefficients. Full article

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

16 pages, 713 KiB

Open AccessFeature PaperArticle

On Fourier Series in the Context of Jacobi Matrices

by José M. A. Matos, Paulo B. Vasconcelos and José A. O. Matos

Axioms 2024, 13(9), 581; https://doi.org/10.3390/axioms13090581 - 27 Aug 2024

Viewed by 531

Abstract

We investigate the properties of matrices that emerge from the application of Fourier series to Jacobi matrices. Specifically, we focus on functions defined by the coefficients of a Fourier series expressed in orthogonal polynomials. In the operational formulation of integro-differential problems, these infinite [...] Read more.

We investigate the properties of matrices that emerge from the application of Fourier series to Jacobi matrices. Specifically, we focus on functions defined by the coefficients of a Fourier series expressed in orthogonal polynomials. In the operational formulation of integro-differential problems, these infinite matrices play a fundamental role. We have derived precise calculation formulas for their elements, enabling exact computation of these operational matrices. Numerical results illustrate the effectiveness of our approach. Full article

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

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21 pages, 331 KiB

Open AccessArticle

Probabilistic and Average Gel’fand Widths of Sobolev Space Equipped with Gaussian Measure in the S_q-Norm

by Ruihuan Wu, Yuqi Liu and Huan Li

Axioms 2024, 13(7), 492; https://doi.org/10.3390/axioms13070492 - 22 Jul 2024

Viewed by 616

Abstract

In this article, we mainly studied the Gel’fand widths of Sobolev space in the probabilistic and average settings. And, we estimated the sharp bounds of the probabilistic Gel’fand

(N, δ)

-widths of multivariate Sobolev space [...] Read more.

In this article, we mainly studied the Gel’fand widths of Sobolev space in the probabilistic and average settings. And, we estimated the sharp bounds of the probabilistic Gel’fand

(N, δ)

-widths of multivariate Sobolev space

M W_{2}^{r} (T^{d})

with mixed derivative equipped with the Gaussian measure in the

S_{q}

-norm by discretization methods. Later, we estimated the sharp bounds of the p-average Gel’fand N-widths of univariate Sobolev space

W_{2}^{r} (T)

and multivariate Sobolev space

M W_{2}^{r} (T^{d})

with mixed derivative equipped with the Gaussian measure in the

S_{q}

-norm. Full article

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

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 601

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)

9 pages, 234 KiB

Open AccessArticle

Asymptotic Conformality and Polygonal Approximation

by Samuel L. Krushkal

Axioms 2024, 13(6), 376; https://doi.org/10.3390/axioms13060376 - 3 Jun 2024

Viewed by 507

Abstract

Univalent functions with asymptotically conformal extension to the boundary form a subclass of functions with quasiconformal extension with rather special features. Such functions arise in various questions of geometric function theory and Teichmüller space theory and have important applications involving conformal and quasiconformal [...] Read more.

Univalent functions with asymptotically conformal extension to the boundary form a subclass of functions with quasiconformal extension with rather special features. Such functions arise in various questions of geometric function theory and Teichmüller space theory and have important applications involving conformal and quasiconformal maps. The paper provides an approximative characterization of local conformality and its connection with univalent polynomials. Also, some other quantitative applications of this connection are given. Full article

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

25 pages, 10343 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Jordan-Type Inequalities and Stratification

by Miloš Mićović and Branko Malešević

Axioms 2024, 13(4), 262; https://doi.org/10.3390/axioms13040262 - 14 Apr 2024

Cited by 1 | Viewed by 1566

Abstract

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding [...] Read more.

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding inequalities through the concept of stratified families of functions. Based on this approach, some optimal approximations of the sinc function are derived by determining the corresponding minimax approximants. Full article

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

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