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Mathematics
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New Trends in Special Functions and Applications
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New Trends in Special Functions and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 15515

Special Issue Editors

Dr. Juan Luis González-Santander

E-Mail Website
Guest Editor
Departamento de Matemáticas. Universidad de Oviedo, Oviedo, Spain
Interests: special functions; mathematical modelling of heat transfer
Special Issues, Collections and Topics in MDPI journals
Dr. Manuel Zamora

E-Mail Website
Guest Editor
Department of Mathematics, University of Oviedo, Oviedo, Spain
Interests: applied mathematics; machine learning; big data

Special Issue Information

Dear Colleagues,

Beyond the well-known elementary functions, special functions gather a great variety of mathematical functions of great importance due to their applications in the fields of physics, mathematics and engineering, and their notation has been consolidated over time.

The main scope of this Special Issue is to publish recent developments in the field of special functions, including classical special functions (gamma, beta, polygamma; higher logarithms; error or incomplete gamma and beta functions; and q-gamma and q-beta functions) or other typical special functions (elliptic integrals; cylinder, hypergeometric and generalized hypergeometric functions; and basic hypergeometric functions). In addition, new results in other types of special functions are welcome, such as Mittag–Leffler function; Riemann and Hurwitz zeta functions; and Apell, Horn and Lauricella functions. This Special Issue is not restricted to the above list and welcomes papers whose content is related to any remarkable property or application in the field of special functions.

Dr. Juan Luis González-Santander
Dr. Manuel Zamora
Guest Editors

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

  • Classical special functions (gamma, beta, error and polygamma functions)
  • Hypergeometric and generalized hypergeometric functions
  • q-gamma and beta functions. Basic hypergeometric functions
  • Cylinder functions (Bessel, Airy and parabolic cylinder functions)
  • Mittag–Leffler function and Riemann and Hurwitz zeta functions

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (7 papers)

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Research

15 pages, 470 KiB  
Article
Two Approximation Formulas for Gamma Function with Monotonic Remainders
by Mansour Mahmoud and Hanan Almuashi
Mathematics 2024, 12(5), 655; https://doi.org/10.3390/math12050655 - 23 Feb 2024
Viewed by 996
Abstract
In this paper, two new approximation formulas with monotonic remainders for the gamma function have been presented. Also, we present some numerical comparisons between our new approximation formulas and some known ones, which demonstrate the superiority of our results. Full article
(This article belongs to the Special Issue New Trends in Special Functions and Applications)
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20 pages, 325 KiB  
Article
Development of a New Zeta Formula and Its Role in Riemann Hypothesis and Quantum Physics
by Saadeldin Abdelaziz, Ahmed Shaker and Mostafa M. Salah
Mathematics 2023, 11(13), 3025; https://doi.org/10.3390/math11133025 - 7 Jul 2023
Cited by 1 | Viewed by 2837
Abstract
In this study, we investigated a new zeta formula in which the zeta function can be expressed as the sum of an infinite series of delta and cosine functions. Our findings demonstrate that this formula possesses duality characteristics and we established a direct [...] Read more.
In this study, we investigated a new zeta formula in which the zeta function can be expressed as the sum of an infinite series of delta and cosine functions. Our findings demonstrate that this formula possesses duality characteristics and we established a direct connection between the Riemann hypothesis and this new formula. Additionally, we explored the behavior of energy or particles in quantum physics within the proposed mathematical model framework based on the new formula. Our model provides a valuable understanding of several important physics inquiries, including the collapse of the wave function during measurement and quantum entanglement, as well as the double slits experiment. Full article
(This article belongs to the Special Issue New Trends in Special Functions and Applications)
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8 pages, 238 KiB  
Article
An Approximation Formula for Nielsen’s Beta Function Involving the Trigamma Function
by Mansour Mahmoud and Hanan Almuashi
Mathematics 2022, 10(24), 4729; https://doi.org/10.3390/math10244729 - 13 Dec 2022
Cited by 1 | Viewed by 1739
Abstract
We prove that the function σ(s) defined by [...] Read more.
We prove that the function σ(s) defined by β(s)=6s2+12s+53s2(2s+3)−ψ′(s)2−σ(s)2s5,s>0, is strictly increasing with the sharp bounds 0<σ(s)<49120, where β(s) is Nielsen’s beta function and ψ′(s) is the trigamma function. Furthermore, we prove that the two functions s↦(−1)1+μβ(s)−6s2+12s+53s2(2s+3)+ψ′(s)2+49μ240s5, μ=0,1 are completely monotonic for s>0. As an application, double inequality for β(s) involving ψ′(s) is obtained, which improve some recent results. Full article
(This article belongs to the Special Issue New Trends in Special Functions and Applications)
14 pages, 298 KiB  
Article
On New Matrix Version Extension of the Incomplete Wright Hypergeometric Functions and Their Fractional Calculus
by Ahmed Bakhet, Abd-Allah Hyder, Areej A. Almoneef, Mohamed Niyaz and Ahmed H. Soliman
Mathematics 2022, 10(22), 4371; https://doi.org/10.3390/math10224371 - 20 Nov 2022
Cited by 4 | Viewed by 1546
Abstract
Through this article, we will discuss a new extension of the incomplete Wright hypergeometric matrix function by using the extended incomplete Pochhammer matrix symbol. First, we give a generalization of the extended incomplete Wright hypergeometric matrix function and state some integral equations and [...] Read more.
Through this article, we will discuss a new extension of the incomplete Wright hypergeometric matrix function by using the extended incomplete Pochhammer matrix symbol. First, we give a generalization of the extended incomplete Wright hypergeometric matrix function and state some integral equations and differential formulas about it. Next, we obtain some results about fractional calculus of these extended incomplete Wright hypergeometric matrix functions. Finally, we discuss an application of the extended incomplete Wright hypergeometric matrix function in the kinetic equations. Full article
(This article belongs to the Special Issue New Trends in Special Functions and Applications)
12 pages, 269 KiB  
Article
Finite and Infinite Hypergeometric Sums Involving the Digamma Function
by Juan Luis González-Santander and Fernando Sánchez Lasheras
Mathematics 2022, 10(16), 2990; https://doi.org/10.3390/math10162990 - 18 Aug 2022
Cited by 2 | Viewed by 1958
Abstract
We calculate some finite and infinite sums containing the digamma function in closed form. For this purpose, we differentiate selected reduction formulas of the hypergeometric function with respect to the parameters applying some derivative formulas of the Pochhammer symbol. Additionally, we compare two [...] Read more.
We calculate some finite and infinite sums containing the digamma function in closed form. For this purpose, we differentiate selected reduction formulas of the hypergeometric function with respect to the parameters applying some derivative formulas of the Pochhammer symbol. Additionally, we compare two different differentiation formulas of the generalized hypergeometric function with respect to the parameters. For some particular cases, we recover some results found in the literature. Finally, all the results have been numerically checked. Full article
(This article belongs to the Special Issue New Trends in Special Functions and Applications)
8 pages, 239 KiB  
Article
Extended Beta and Gamma Matrix Functions via 2-Parameter Mittag-Leffler Matrix Function
by Rahul Goyal, Praveen Agarwal, Georgia Irina Oros and Shilpi Jain
Mathematics 2022, 10(6), 892; https://doi.org/10.3390/math10060892 - 11 Mar 2022
Cited by 12 | Viewed by 2377
Abstract
The main aim of this article is to study an extension of the Beta and Gamma matrix functions by using a two-parameter Mittag-Leffler matrix function. In particular, we investigate certain properties of these extended matrix functions such as symmetric relation, integral representations, summation [...] Read more.
The main aim of this article is to study an extension of the Beta and Gamma matrix functions by using a two-parameter Mittag-Leffler matrix function. In particular, we investigate certain properties of these extended matrix functions such as symmetric relation, integral representations, summation relations, generating relation and functional relation. Full article
(This article belongs to the Special Issue New Trends in Special Functions and Applications)
6 pages, 251 KiB  
Article
A Note on Some Reduction Formulas for the Incomplete Beta Function and the Lerch Transcendent
by Juan Luis González-Santander
Mathematics 2021, 9(13), 1486; https://doi.org/10.3390/math9131486 - 24 Jun 2021
Cited by 3 | Viewed by 2376
Abstract
We derive new reduction formulas for the incomplete beta function Bν,0,z and the Lerch transcendent Φz,1,ν in terms of elementary functions when ν is rational and z is complex. As an application, we [...] Read more.
We derive new reduction formulas for the incomplete beta function Bν,0,z and the Lerch transcendent Φz,1,ν in terms of elementary functions when ν is rational and z is complex. As an application, we calculate some new integrals. Additionally, we use these reduction formulas to test the performance of the algorithms devoted to the numerical evaluation of the incomplete beta function. Full article
(This article belongs to the Special Issue New Trends in Special Functions and Applications)
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