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Special Issue "New Trends in Special Functions and Applications"

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

Deadline for manuscript submissions: 31 December 2023 | Viewed by 6303

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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

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

Published Papers (6 papers)

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Research

Open AccessArticle

Development of a New Zeta Formula and Its Role in Riemann Hypothesis and Quantum Physics

Saadeldin Abdelaziz

Ahmed Shaker

and

Mostafa M. Salah

Mathematics 2023, 11(13), 3025; https://doi.org/10.3390/math11133025 - 07 Jul 2023

Viewed by 550

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 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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Open AccessArticle

An Approximation Formula for Nielsen’s Beta Function Involving the Trigamma Function

Mansour Mahmoud

and

Hanan Almuashi

Mathematics 2022, 10(24), 4729; https://doi.org/10.3390/math10244729 - 13 Dec 2022

Viewed by 724

Abstract

We prove that the function

σ (s)

defined by [...] Read more.

We prove that the function

σ (s)

defined by

β (s) = \frac{6 s^{2} + 12 s + 5}{3 s^{2} (2 s + 3)} - \frac{ψ^{'} (s)}{2} - \frac{σ (s)}{2 s^{5}}, s > 0

, is strictly increasing with the sharp bounds

0 < σ (s) < \frac{49}{120}

, where

β (s)

is Nielsen’s beta function and

ψ^{'} (s)

is the trigamma function. Furthermore, we prove that the two functions

s \mapsto {(- 1)}^{1 + μ} [β (s) - \frac{6 s^{2} + 12 s + 5}{3 s^{2} (2 s + 3)} + \frac{ψ^{'} (s)}{2} + \frac{49 μ}{240 s^{5}}], 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)

Open AccessArticle

On New Matrix Version Extension of the Incomplete Wright Hypergeometric Functions and Their Fractional Calculus

and

Mathematics 2022, 10(22), 4371; https://doi.org/10.3390/math10224371 - 20 Nov 2022

Viewed by 781

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 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)

Open AccessFeature PaperArticle

Finite and Infinite Hypergeometric Sums Involving the Digamma Function

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 812

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 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)

Open AccessArticle

Extended Beta and Gamma Matrix Functions via 2-Parameter Mittag-Leffler Matrix Function

and

Mathematics 2022, 10(6), 892; https://doi.org/10.3390/math10060892 - 11 Mar 2022

Cited by 8 | Viewed by 1336

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.

(This article belongs to the Special Issue New Trends in Special Functions and Applications)

Open AccessArticle

A Note on Some Reduction Formulas for the Incomplete Beta Function and the Lerch Transcendent

Juan Luis González-Santander

Mathematics 2021, 9(13), 1486; https://doi.org/10.3390/math9131486 - 24 Jun 2021

Cited by 2 | Viewed by 1164

Abstract

We derive new reduction formulas for the incomplete beta function

B (ν, 0, z)

and the Lerch transcendent

Φ (z, 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

B (ν, 0, z)

and the Lerch transcendent

Φ (z, 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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