Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
3.5
2.4
Journals
Mathematics
Special Issues
Special Functions with Applications to Mathematical Physics
share announcement

Special Functions with Applications to Mathematical Physics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 52437

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Francesco Mainardi

E-Mail Website
Guest Editor
Department of Physics and Astronomy, University of Bologna, Via Irnerio, 46, I-40126 Bologna, Italy
Interests: special functions; fractional calculus complex analysis; asymptotic methods; diffusion and wave propagation problems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue includes theories and applications of high transcendental functions mainly included in the list of keywords:

  • Mittag–Leffler and related functions, and their applications in mathematical physics;
  • Wright and related functions and their applications in mathematical physics;
  • Exponential Integrals and their extensions with applications in mathematical physics;
  • Generalized hypergeometric functions and their extensions with applications.

However, this Special Issue is not limited to the above list, when the content of a paper is clearly related to some high transcendental functions and their applications. Special attention is reserved for the special functions exhibiting some relevance in the framework of the theories and applications of the fractional calculus and in their visualization through illuminating plots. Both research and survey pages are well accepted.

Prof. Francesco Mainardi
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Mittag–Leffler and related functions, and their applications in mathematical physics
  • Wright and related functions and their applications in mathematical physics
  • Exponential integrals and their extensions with applications in mathematical physics
  • Generalized hypergeometric functions and their extensions with applications

Published Papers (23 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

Jump to: Review, Other

34 pages, 577 KiB  
Article
The Integral Mittag-Leffler, Whittaker and Wright Functions
by Alexander Apelblat and Juan Luis González-Santander
Mathematics 2021, 9(24), 3255; https://doi.org/10.3390/math9243255 - 15 Dec 2021
Cited by 9 | Viewed by 1824
Abstract
Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In most reported cases, these new integral functions are [...] Read more.
Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In most reported cases, these new integral functions are expressed as generalized hypergeometric functions but also in terms of elementary and special functions. The behavior of some of the new integral functions is presented in graphical form. By using the MATHEMATICA program to obtain infinite sums that define the Mittag-Leffler, Whittaker, and Wright functions and also their corresponding integral functions, these functions and many new Laplace transforms of them are also reported in the Appendices for integral and fractional values of parameters. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
►▼ Show Figures

Figure 1

13 pages, 1927 KiB  
Article
Mathematical Description and Laboratory Study of Electrophysical Methods of Localization of Geodeformational Changes during the Control of the Railway Roadbed
by Artem Bykov, Anastasia Grecheneva, Oleg Kuzichkin, Dmitry Surzhik, Gleb Vasilyev and Yerbol Yerbayev
Mathematics 2021, 9(24), 3164; https://doi.org/10.3390/math9243164 - 08 Dec 2021
Cited by 5 | Viewed by 2041
Abstract
Currently, the load on railway tracks is increasing due to the increase in freight traffic. Accordingly, more and more serious requirements are being imposed on the reliability of the roadbed, which means that studies of methods for monitoring the integrity of the railway [...] Read more.
Currently, the load on railway tracks is increasing due to the increase in freight traffic. Accordingly, more and more serious requirements are being imposed on the reliability of the roadbed, which means that studies of methods for monitoring the integrity of the railway roadbed are relevant. The article provides a mathematical substantiation of the possibility of using seismoelectric and phasemetric methods of geoelectric control of the roadbed of railway tracks in order to identify defects and deformations at an early stage of their occurrence. The methods of laboratory modeling of the natural–technical system “railway track” are considered in order to assess the prospects of using the presented methods. The results of laboratory studies are presented, which have shown their high efficiency in registering a weak useful electrical signal caused by seismoacoustic effects against the background of high-level external industrial and natural interference. In the course of laboratory modeling, it was found that on the amplitude spectra of the output electrical signals of the investigated geological medium in the presence of an elastic harmonic action with a frequency of 70 Hz, the frequency of a harmonic electrical signal with a frequency of 40 Hz is observed. In laboratory modeling, phase images were obtained for the receiving line when simulating the process of sinking the soil base of the railway bed, confirming the presence of a transient process that causes a shift in the initial phase of the signal Δφ = 40° by ~45° (Δφ’ = 85°), which allows detection of the initial stage of failure formation. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
►▼ Show Figures

Figure 1

12 pages, 271 KiB  
Article
Basic Fundamental Formulas for Wiener Transforms Associated with a Pair of Operators on Hilbert Space
by Hyun Soo Chung
Mathematics 2021, 9(21), 2738; https://doi.org/10.3390/math9212738 - 28 Oct 2021
Cited by 1 | Viewed by 872
Abstract
Segal introduce the Fourier–Wiener transform for the class of polynomial cylinder functions on Hilbert space, and Hida then develop this concept. Negrin define the extended Wiener transform with Hayker et al. In recent papers, Hayker et al. establish the existence, the composition formula, [...] Read more.
Segal introduce the Fourier–Wiener transform for the class of polynomial cylinder functions on Hilbert space, and Hida then develop this concept. Negrin define the extended Wiener transform with Hayker et al. In recent papers, Hayker et al. establish the existence, the composition formula, the inversion formula, and the Parseval relation for the Wiener transform. But, they do not establish homomorphism properties for the Wiener transform. In this paper, the author establishes some basic fundamental formulas for the Wiener transform via some concepts and motivations introduced by Segal and used by Hayker et al. We then state the usefulness of basic fundamental formulas as some applications. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
15 pages, 296 KiB  
Article
Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications
by Yuri Luchko
Mathematics 2021, 9(17), 2132; https://doi.org/10.3390/math9172132 - 02 Sep 2021
Cited by 20 | Viewed by 2083
Abstract
In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely [...] Read more.
In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
16 pages, 297 KiB  
Article
Fractional Calculus in Russia at the End of XIX Century
by Sergei Rogosin and Maryna Dubatovskaya
Mathematics 2021, 9(15), 1736; https://doi.org/10.3390/math9151736 - 22 Jul 2021
Cited by 7 | Viewed by 1719
Abstract
In this survey paper, we analyze the development of Fractional Calculus in Russia at the end of the XIX century, in particular, the results by A. V. Letnikov, N. Ya. Sonine, and P. A. Nekrasov. Some of the discussed results are either unknown [...] Read more.
In this survey paper, we analyze the development of Fractional Calculus in Russia at the end of the XIX century, in particular, the results by A. V. Letnikov, N. Ya. Sonine, and P. A. Nekrasov. Some of the discussed results are either unknown or inaccessible. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
10 pages, 415 KiB  
Article
The Asymptotic Expansion of a Function Introduced by L.L. Karasheva
by Richard Paris
Mathematics 2021, 9(12), 1454; https://doi.org/10.3390/math9121454 - 21 Jun 2021
Viewed by 1335
Abstract
The asymptotic expansion for x→±∞ of the entire function [...] Read more.
The asymptotic expansion for x→±∞ of the entire function Fn,σ(x;μ)=∑k=0∞sin(nγk)sinγkxkk!Γ(μ−σk),γk=(k+1)π2n for μ>0, 0<σ<1 and n=1,2,… is considered. In the special case σ=α/(2n), with 0<α<1, this function was recently introduced by L.L. Karasheva (J. Math. Sciences, 250 (2020) 753–759) as a solution of a fractional-order partial differential equation. By expressing Fn,σ(x;μ) as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This was found to depend critically on the parameter σ (and to a lesser extent on the integer n). Numerical results are presented to illustrate the accuracy of the different expansions obtained. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
►▼ Show Figures

Figure 1

12 pages, 2889 KiB  
Article
On the Multistage Differential Transformation Method for Analyzing Damping Duffing Oscillator and Its Applications to Plasma Physics
by Noufe H. Aljahdaly and S. A. El-Tantawy
Mathematics 2021, 9(4), 432; https://doi.org/10.3390/math9040432 - 22 Feb 2021
Cited by 48 | Viewed by 2169
Abstract
The multistage differential transformation method (MSDTM) is used to find an approximate solution to the forced damping Duffing equation (FDDE). In this paper, we prove that the MSDTM can predict the solution in the long domain as compared to differential transformation method (DTM) [...] Read more.
The multistage differential transformation method (MSDTM) is used to find an approximate solution to the forced damping Duffing equation (FDDE). In this paper, we prove that the MSDTM can predict the solution in the long domain as compared to differential transformation method (DTM) and more accurately than the modified differential transformation method (MDTM). In addition, the maximum residual errors for DTM and its modification methods (MSDTM and MDTM) are estimated. As a real application to the obtained solution, we investigate the oscillations in a complex unmagnetized plasma. To do that, the fluid govern equations of plasma species is reduced to the modified Korteweg–de Vries–Burgers (mKdVB) equation. After that, by using a suitable transformation, the mKdVB equation is transformed into the forced damping Duffing equation. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
►▼ Show Figures

Figure 1

24 pages, 388 KiB  
Article
Some Properties of the Kilbas-Saigo Function
by Lotfi Boudabsa and Thomas Simon
Mathematics 2021, 9(3), 217; https://doi.org/10.3390/math9030217 - 22 Jan 2021
Cited by 18 | Viewed by 1864
Abstract
We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient [...] Read more.
We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
12 pages, 337 KiB  
Article
Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals
by Árpád Baricz, Dragana Jankov Maširević and Tibor K. Pogány
Mathematics 2021, 9(2), 129; https://doi.org/10.3390/math9020129 - 08 Jan 2021
Cited by 3 | Viewed by 3395
Abstract
The cumulative distribution function of the non-central chi-square distribution χn′2(λ) of n degrees of freedom possesses an integral representation. Here we rewrite this integral in terms of a lower incomplete gamma function applying two of the second [...] Read more.
The cumulative distribution function of the non-central chi-square distribution χn′2(λ) of n degrees of freedom possesses an integral representation. Here we rewrite this integral in terms of a lower incomplete gamma function applying two of the second mean-value theorems for definite integrals, which are of Bonnet type and Okamura’s variant of the du Bois–Reymond theorem. Related results are exposed concerning the small argument cases in cumulative distribution function (CDF) and their asymptotic behavior near the origin. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
16 pages, 310 KiB  
Article
Some Relationships for the Generalized Integral Transform on Function Space
by Hyun Soo Chung
Mathematics 2020, 8(12), 2246; https://doi.org/10.3390/math8122246 - 19 Dec 2020
Viewed by 1496
Abstract
In this paper, we recall a more generalized integral transform, a generalized convolution product and a generalized first variation on function space. The Gaussian process and the bounded linear operators on function space are used to define them. We then establish the existence [...] Read more.
In this paper, we recall a more generalized integral transform, a generalized convolution product and a generalized first variation on function space. The Gaussian process and the bounded linear operators on function space are used to define them. We then establish the existence and various relationships between the generalized integral transform and the generalized convolution product. Furthermore, we obtain some relationships between the generalized integral transform and the generalized first variation with the generalized Cameron–Storvick theorem. Finally, some applications are demonstrated as examples. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
17 pages, 1245 KiB  
Article
Laguerre-Type Exponentials, Laguerre Derivatives and Applications. A Survey
by Paolo Emilio Ricci
Mathematics 2020, 8(11), 2054; https://doi.org/10.3390/math8112054 - 18 Nov 2020
Cited by 10 | Viewed by 1838
Abstract
Laguerrian derivatives and related autofunctions are presented that allow building new special functions determined by the action of a differential isomorphism within the space of analytical functions. Such isomorphism can be iterated every time, so that the resulting construction can be re-submitted endlessly [...] Read more.
Laguerrian derivatives and related autofunctions are presented that allow building new special functions determined by the action of a differential isomorphism within the space of analytical functions. Such isomorphism can be iterated every time, so that the resulting construction can be re-submitted endlessly in a cyclic way. Some applications of this theory are made in the field of population dynamics and in the solution of Cauchy’s problems for particular linear dynamical systems. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
►▼ Show Figures

Figure 1

21 pages, 415 KiB  
Article
Transformations of the Hypergeometric 4F3 with One Unit Shift: A Group Theoretic Study
by Dmitrii Karp and Elena Prilepkina
Mathematics 2020, 8(11), 1966; https://doi.org/10.3390/math8111966 - 05 Nov 2020
Cited by 4 | Viewed by 1586
Abstract
We study the group of transformations of 4F3 hypergeometric functions evaluated at unity with one unit shift in parameters. We reveal the general form of this family of transformations and its group property. Next, we use explicitly known transformations to generate [...] Read more.
We study the group of transformations of 4F3 hypergeometric functions evaluated at unity with one unit shift in parameters. We reveal the general form of this family of transformations and its group property. Next, we use explicitly known transformations to generate a subgroup whose structure is then thoroughly studied. Using some known results for 3F2 transformation groups, we show that this subgroup is isomorphic to the direct product of the symmetric group of degree 5 and 5-dimensional integer lattice. We investigate the relation between two-term 4F3 transformations from our group and three-term 3F2 transformations and present a method for computing the coefficients of the contiguous relations for 3F2 functions evaluated at unity. We further furnish a class of summation formulas associated with the elements of our group. In the appendix to this paper, we give a collection of Wolfram Mathematica® routines facilitating the group calculations. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
22 pages, 951 KiB  
Article
Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach
by Alexander Apelblat
Mathematics 2020, 8(5), 657; https://doi.org/10.3390/math8050657 - 26 Apr 2020
Cited by 19 | Viewed by 5674
Abstract
In this work, properties of one- or two-parameter Mittag-Leffler functions are derived using the Laplace transform approach. It is demonstrated that manipulations with the pair direct–inverse transform makes it far more easy than previous methods to derive known and new properties of the [...] Read more.
In this work, properties of one- or two-parameter Mittag-Leffler functions are derived using the Laplace transform approach. It is demonstrated that manipulations with the pair direct–inverse transform makes it far more easy than previous methods to derive known and new properties of the Mittag-Leffler functions. Moreover, it is shown that sums of infinite series of the Mittag-Leffler functions can be expressed as convolution integrals, while the derivatives of the Mittag-Leffler functions with respect to their parameters are expressible as double convolution integrals. The derivatives can also be obtained from integral representations of the Mittag-Leffler functions. On the other hand, direct differentiation of the Mittag-Leffler functions with respect to parameters produces an infinite power series, whose coefficients are quotients of the digamma and gamma functions. Closed forms of these series can be derived when the parameters are set to be integers. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
►▼ Show Figures

Figure 1

18 pages, 308 KiB  
Article
Mathematical Aspects of Krätzel Integral and Krätzel Transform
by Arak M. Mathai and Hans J. Haubold
Mathematics 2020, 8(4), 526; https://doi.org/10.3390/math8040526 - 03 Apr 2020
Cited by 5 | Viewed by 1682
Abstract
A real scalar variable integral is known in the literature by different names in different disciplines. It is basically a Bessel integral called specifically Krätzel integral. An integral transform with this Krätzel function as kernel is known as Krätzel transform. This article examines [...] Read more.
A real scalar variable integral is known in the literature by different names in different disciplines. It is basically a Bessel integral called specifically Krätzel integral. An integral transform with this Krätzel function as kernel is known as Krätzel transform. This article examines some mathematical properties of Krätzel integral, its connection to Mellin convolutions and statistical distributions, its computable representations, and its extensions to multivariate and matrix-variate cases, in both the real and complex domains. An extension in the pathway family of functions is also explored. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
13 pages, 357 KiB  
Article
Asymptotic Expansion of the Modified Exponential Integral Involving the Mittag-Leffler Function
by Richard Paris
Mathematics 2020, 8(3), 428; https://doi.org/10.3390/math8030428 - 16 Mar 2020
Cited by 4 | Viewed by 2607
Abstract
We consider the asymptotic expansion of the generalised exponential integral involving the Mittag-Leffler function introduced recently by Mainardi and Masina [Fract. Calc. Appl. Anal. 21 (2018) 1156–1169]. We extend the definition of this function using the two-parameter Mittag-Leffler function. The expansions of [...] Read more.
We consider the asymptotic expansion of the generalised exponential integral involving the Mittag-Leffler function introduced recently by Mainardi and Masina [Fract. Calc. Appl. Anal. 21 (2018) 1156–1169]. We extend the definition of this function using the two-parameter Mittag-Leffler function. The expansions of the similarly extended sine and cosine integrals are also discussed. Numerical examples are presented to illustrate the accuracy of each type of expansion obtained. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
►▼ Show Figures

Figure 1

Review

Jump to: Research, Other

15 pages, 338 KiB  
Review
Series in Le Roy Type Functions: A Set of Results in the Complex Plane—A Survey
by Jordanka Paneva-Konovska
Mathematics 2021, 9(12), 1361; https://doi.org/10.3390/math9121361 - 12 Jun 2021
Cited by 8 | Viewed by 1366
Abstract
This study is based on a part of the results obtained in the author’s publications. An enumerable family of the Le Roy type functions is considered herein. The asymptotic formula for these special functions in the cases of ‘large’ values of indices, that [...] Read more.
This study is based on a part of the results obtained in the author’s publications. An enumerable family of the Le Roy type functions is considered herein. The asymptotic formula for these special functions in the cases of ‘large’ values of indices, that has been previously obtained, is provided. Further, series defined by means of the Le Roy type functions are considered. These series are studied in the complex plane. Their domains of convergence are given and their behaviour is investigated ‘near’ the boundaries of the domains of convergence. The discussed asymptotic formula is used in the proofs of the convergence theorems for the considered series. A theorem of the Cauchy–Hadamard type is provided. Results of Abel, Tauber and Littlewood type, which are analogues to the corresponding theorems for the classical power series, are also proved. At last, various interesting particular cases of the discussed special functions are considered. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
27 pages, 4999 KiB  
Review
The Bateman Functions Revisited after 90 Years—A Survey of Old and New Results
by Alexander Apelblat, Armando Consiglio and Francesco Mainardi
Mathematics 2021, 9(11), 1273; https://doi.org/10.3390/math9111273 - 01 Jun 2021
Cited by 1 | Viewed by 2650
Abstract
The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as [...] Read more.
The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions, their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to generalizations of these functions and to the Bateman-integral function known in the literature. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
►▼ Show Figures

Figure 1

14 pages, 309 KiB  
Review
Some Applications of the Wright Function in Continuum Physics: A Survey
by Yuriy Povstenko
Mathematics 2021, 9(2), 198; https://doi.org/10.3390/math9020198 - 19 Jan 2021
Cited by 5 | Viewed by 1726
Abstract
The Wright function is a generalization of the exponential function and the Bessel functions. Integral relations between the Mittag–Leffler functions and the Wright function are presented. The applications of the Wright function and the Mainardi function to description of diffusion, heat conduction, thermal [...] Read more.
The Wright function is a generalization of the exponential function and the Bessel functions. Integral relations between the Mittag–Leffler functions and the Wright function are presented. The applications of the Wright function and the Mainardi function to description of diffusion, heat conduction, thermal and diffusive stresses, and nonlocal elasticity in the framework of fractional calculus are discussed. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
40 pages, 643 KiB  
Review
A Guide to Special Functions in Fractional Calculus
by Virginia Kiryakova
Mathematics 2021, 9(1), 106; https://doi.org/10.3390/math9010106 - 05 Jan 2021
Cited by 34 | Viewed by 4068
Abstract
Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as [...] Read more.
Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
28 pages, 509 KiB  
Review
Exact Values of the Gamma Function from Stirling’s Formula
by Victor Kowalenko
Mathematics 2020, 8(7), 1058; https://doi.org/10.3390/math8071058 - 01 Jul 2020
Cited by 1 | Viewed by 3618 | Correction
Abstract
In this work the complete version of Stirling’s formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma function over all branches of the complex plane. Exact values [...] Read more.
In this work the complete version of Stirling’s formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma function over all branches of the complex plane. Exact values can only be obtained by regularization. Two methods are introduced: Borel summation and Mellin–Barnes (MB) regularization. The Borel-summed remainder is composed of an infinite convergent sum of exponential integrals and discontinuous logarithmic terms that emerge in specific sectors and on lines known as Stokes sectors and lines, while the MB-regularized remainders reduce to one complex MB integral with similar logarithmic terms. As a result that the domains of convergence overlap, two MB-regularized asymptotic forms can often be used to evaluate the logarithm of the gamma function. Though the Borel-summed remainder has to be truncated, it is found that both remainders when summed with (1) the truncated asymptotic series, (2) Stirling’s formula and (3) the logarithmic terms arising from the higher branches of the complex plane yield identical values for the logarithm of the gamma function. Where possible, they also agree with results from Mathematica. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
►▼ Show Figures

Figure 1

16 pages, 320 KiB  
Review
The Four-Parameters Wright Function of the Second kind and its Applications in FC
by Yuri Luchko
Mathematics 2020, 8(6), 970; https://doi.org/10.3390/math8060970 - 12 Jun 2020
Cited by 7 | Viewed by 2133
Abstract
In this survey paper, we present both some basic properties of the four-parameters Wright function and its applications in Fractional Calculus. For applications in Fractional Calculus, the four-parameters Wright function of the second kind is especially important. In the paper, three case studies [...] Read more.
In this survey paper, we present both some basic properties of the four-parameters Wright function and its applications in Fractional Calculus. For applications in Fractional Calculus, the four-parameters Wright function of the second kind is especially important. In the paper, three case studies illustrating a wide spectrum of its applications are presented. The first case study deals with the scale-invariant solutions to a one-dimensional time-fractional diffusion-wave equation that can be represented in terms of the Wright function of the second kind and the four-parameters Wright function of the second kind. In the second case study, we consider a subordination formula for the solutions to a multi-dimensional space-time-fractional diffusion equation with different orders of the fractional derivatives. The kernel of the subordination integral is a special case of the four-parameters Wright function of the second kind. Finally, in the third case study, we shortly present an application of an operational calculus for a composed Erdélyi-Kober fractional operator for solving some initial-value problems for the fractional differential equations with the left- and right-hand sided Erdélyi-Kober fractional derivatives. In particular, we present an example with an explicit solution in terms of the four-parameters Wright function of the second kind. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
26 pages, 4202 KiB  
Review
The Wright Functions of the Second Kind in Mathematical Physics
by Francesco Mainardi and Armando Consiglio
Mathematics 2020, 8(6), 884; https://doi.org/10.3390/math8060884 - 01 Jun 2020
Cited by 27 | Viewed by 3441
Abstract
In this review paper, we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics. We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We [...] Read more.
In this review paper, we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics. We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices, we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
►▼ Show Figures

Figure 1

Other

Jump to: Research, Review

1 pages, 178 KiB  
Correction
Correction: Kowalenko, V. Exact Values of the Gamma Function from Stirling’s Formula. Mathematics 2020, 8, 1058
by Victor Kowalenko
Mathematics 2023, 11(9), 2151; https://doi.org/10.3390/math11092151 - 04 May 2023
Viewed by 471
Abstract
In the original publication [...] Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-23
Back to TopTop