Integral Transforms and Special Functions in Applied Mathematics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: 31 May 2024 | Viewed by 4349

Special Issue Editors

Department of Statistics, Informatics and Mathematics, Public University of Navarra, 31008 Pamplona, Spain
Interests: approximation theory, in particular asymptotic approximation; special functions
Departement of Mathematics, Holon Institute of Technology, Holon 5810201, Israel
Interests: special functions; function theory; inequalities
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Special functions remain an active field of study even after several centuries of history, appearing in a variety of ways, such as solutions of differential/integral equations, densities of probability distributions, counting numbers of combinatorial objects, number-theoretic quantities, in evaluation of integrals and summation of series, as solutions of the Riemann–Hilbert problems, as explicit expressions for (multiple) orthogonal polynomials, in fractional calculus, and as matrix elements of group representations, to name a few. Integral transforms (such as Fourier, Laplace, Mellin, and Hankel, among others) are a powerful and versatile tool with applications in a wide range of fields, including pure mathematics, physics, engineering, and computer science, being indispensable in functional analysis, ODEs and PDEs, asymptotic analysis, signal processing and compression, control systems and cryptography. A large database of integral transforms of special functions is known, and the two are often related and can interact in several ways: special functions can be defined via integral transforms, serve as kernels of integral transforms and be used for inverting integral transforms.

This Special Issue will be devoted to both the theory and applications of special functions, especially those that can be represented by means of an integral transform or that arise in the study of integral transforms. We invite papers on topics in both mathematics and sciences, where special functions and integral transforms play an important role. Although the main emphasis will be on applications, i.e., the use of special functions or integral transforms in any scientific discipline, papers related to the fundamental theory, such as asymptotics, symmetries, representations, transformation and summation formulas, approximations, inequalities, zeros, monotonicity and complex-analytic properties, are also welcome.

Prof. Dr. Jose Luis Lopez
Dr. Dmitrii Karp
Guest Editors

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Keywords

  • integral transforms
  • special functions
  • orthogonal polynomials
  • applied mathematics
  • asymptotic approximation
  • analytic representation
  • inequalities

Published Papers (5 papers)

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Research

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20 pages, 315 KiB  
Article
Novel Formulas for B-Splines, Bernstein Basis Functions, and Special Numbers: Approach to Derivative and Functional Equations of Generating Functions
by Yilmaz Simsek
Mathematics 2024, 12(1), 65; https://doi.org/10.3390/math12010065 - 24 Dec 2023
Cited by 1 | Viewed by 495
Abstract
The purpose of this article is to give relations among the uniform B-splines, the Bernstein basis functions, and certain families of special numbers and polynomials with the aid of the generating functions method. We derive a relation between generating functions for the [...] Read more.
The purpose of this article is to give relations among the uniform B-splines, the Bernstein basis functions, and certain families of special numbers and polynomials with the aid of the generating functions method. We derive a relation between generating functions for the uniform B-splines and generating functions for the Bernstein basis functions. We derive some functional equations for these generating functions. Using the higher-order partial derivative equations of these generating functions, we derive both the generalized de Boor recursion relation and the higher-order derivative formula of uniform B-splines in terms of Bernstein basis functions. Using the functional equations of these generating functions, we derive the relations among the Bernstein basis functions, the uniform B-splines, the Apostol-Bernoulli numbers and polynomials, the Aposto–Euler numbers and polynomials, the Eulerian numbers and polynomials, and the Stirling numbers. Applying the p-adic integrals to these polynomials, we derive many novel formulas. Furthermore, by applying the Laplace transformation to these generating functions, we derive infinite series representations for the uniform B-splines and the Bernstein basis functions. Full article
(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)
17 pages, 319 KiB  
Article
Bohr’s Phenomenon for the Solution of Second-Order Differential Equations
by Saiful R. Mondal
Mathematics 2024, 12(1), 39; https://doi.org/10.3390/math12010039 - 22 Dec 2023
Viewed by 903
Abstract
The aim of this work is to establish a connection between Bohr’s radius and the analytic and normalized solutions of two differential second-order differential equations, namely [...] Read more.
The aim of this work is to establish a connection between Bohr’s radius and the analytic and normalized solutions of two differential second-order differential equations, namely y(z)+a(z)y(z)+b(z)y(z)=0 and z2y(z)+a(z)y(z)+b(z)y(z)=d(z). Using differential subordination, we find the upper bound of the Bohr and Rogosinski radii of the normalized solution F(z) of the above differential equations. We construct several examples by judicious choice of a(z), b(z) and d(z). The examples include several special functions like Airy functions, classical and generalized Bessel functions, error functions, confluent hypergeometric functions and associate Laguerre polynomials. Full article
(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)
10 pages, 271 KiB  
Article
Integral Representations of a Generalized Linear Hermite Functional
by Roberto S. Costas-Santos
Mathematics 2023, 11(14), 3227; https://doi.org/10.3390/math11143227 - 22 Jul 2023
Viewed by 543
Abstract
In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given. Full article
(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)
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14 pages, 560 KiB  
Article
Asymptotic Expansions for Moench’s Integral Transform of Hydrology
by José L. López, Pedro Pagola and Ester Pérez Sinusía
Mathematics 2023, 11(14), 3053; https://doi.org/10.3390/math11143053 - 10 Jul 2023
Viewed by 1225
Abstract
Theis’ theory (1935), later improved by Hantush & Jacob (1955) and Moench (1971), is a technique designed to study the water level in aquifers. The key formula in this theory is a certain integral transform [...] Read more.
Theis’ theory (1935), later improved by Hantush & Jacob (1955) and Moench (1971), is a technique designed to study the water level in aquifers. The key formula in this theory is a certain integral transform H[g](r,t) of the pumping function g that depends on the time t and the relative position r to the pumping point as well as on other physical parameters. Several analytic approximations of H[g](r,t) have been investigated in the literature that are valid and accurate in certain regions of r, t and the mentioned physical parameters. In this paper, the analysis of possible analytic approximations of H[g](r,t) is completed by investigating asymptotic expansions of H[g](r,t) in a region of the parameters that is of interest in practical situations, but that has not yet been investigated. Explicit and/or recursive algorithms for the computation of the coefficients of the expansions and estimates for the remainders are provided. Some numerical examples based on an actual physical experiment conducted by Layne-Western Company in 1953 illustrate the accuracy of the approximations. Full article
(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)
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Review

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39 pages, 570 KiB  
Review
Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey
by Virginia Kiryakova and Jordanka Paneva-Konovska
Mathematics 2024, 12(2), 319; https://doi.org/10.3390/math12020319 - 18 Jan 2024
Viewed by 582
Abstract
In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions pΨq [...] Read more.
In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions pΨq and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and pFq-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the H¯-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and H¯-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of Γ-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and H¯-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems. Full article
(This article belongs to the Special Issue Integral Transforms and Special Functions in Applied Mathematics)
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