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Mathematics
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Special Functions with Applications
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Special Functions with Applications

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 30 November 2025 | Viewed by 1179

Special Issue Editor

Dr. Vito Lampret

E-Mail Website
Guest Editor
Faculty of Civil and Geodetic Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia
Interests: asymptotic expansion; completely monotonic function; gamma function; elliptic integral; trigonometric function; arithmetic mean

Special Issue Information

Dear Colleagues,

This Special Issue, “Special Functions with Applications”, focuses on the study and utilization of special functions, which are mathematically significant and arise in applied mathematics and various fields of science. These functions, such as Bessel functions, hypergeometric functions, orthogonal polynomials, functions relating to the gamma function, etc., play a crucial role in solving differential equations, integral transforms, and problems in mathematical physics. The aim of this topic is to explore the theoretical foundations, computational methods, and practical applications of special functions, bridging the gap between pure mathematics and real-world problems.

This topic welcomes research on the development of new analytical and numerical techniques for evaluating special functions, as well as their applications in areas such as quantum mechanics, signal processing, fluid dynamics, and statistical mechanics. Contributions may also include interdisciplinary studies that leverage special functions to solve complex problems in engineering, computer science, and beyond. By fostering collaboration between mathematicians, physicists, and engineers, this Special Issue seeks to advance our understanding of special functions and expand their utility in cutting-edge applications.

Dr. Vito Lampret
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

  • special functions
  • Airy function
  • Bessel function
  • Fresnel, Hankel, Kelvin, Lambert, Mathieu, etc., functions
  • hypergeometric functions
  • orthogonal polynomials Functions relating to the gamma function (fractional calculus functions)
  • differential equations
  • integral transforms
  • mathematical physics computational methods
  • quantum mechanics
  • signal processing
  • fluid dynamics
  • statistical mechanics
  • applied mathematics
  • numerical techniques

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

Published Papers (2 papers)

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Research

20 pages, 2586 KiB  
Article
An In-Depth Investigation of the Riemann Zeta Function Using Infinite Numbers
by Emmanuel Thalassinakis
Mathematics 2025, 13(9), 1483; https://doi.org/10.3390/math13091483 - 30 Apr 2025
Viewed by 697
Abstract
This study focuses on an in-depth investigation of the Riemann zeta function. For this purpose, infinite numbers and rotational infinite numbers, which have been introduced in previous studies published by the author, are used. These numbers are a powerful tool for solving problems [...] Read more.
This study focuses on an in-depth investigation of the Riemann zeta function. For this purpose, infinite numbers and rotational infinite numbers, which have been introduced in previous studies published by the author, are used. These numbers are a powerful tool for solving problems involving infinity that are otherwise difficult to solve. Infinite numbers are a superset of complex numbers and can be either complex numbers or some quantification of infinity. The Riemann zeta function can be written as a sum of three rotational infinite numbers, each of which represents infinity. Using these infinite numbers and their properties, a correlation of the non-trivial zeros of the Riemann zeta function with each other is revealed and proven. In addition, an interesting relation between the Euler–Mascheroni constant (γ) and the non-trivial zeros of the Riemann zeta function is proven. Based on this analysis, complex series limits are calculated and important conclusions about the Riemann zeta function are drawn. It turns out that when we have non-trivial zeros of the Riemann zeta function, the corresponding Dirichlet series increases linearly, in contrast to the other cases where this series also includes a fluctuating term. The above theoretical results are fully verified using numerical computations. Furthermore, a new numerical method is presented for calculating the non-trivial zeros of the Riemann zeta function, which lie on the critical line. In summary, by using infinite numbers, aspects of the Riemann zeta function are explored and revealed from a different perspective; additionally, interesting mathematical relationships that are difficult or impossible to solve with other methods are easily analyzed and solved. Full article
(This article belongs to the Special Issue Special Functions with Applications)
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30 pages, 2164 KiB  
Article
More Theory About Infinite Numbers and Important Applications
by Emmanuel Thalassinakis
Mathematics 2025, 13(9), 1390; https://doi.org/10.3390/math13091390 - 24 Apr 2025
Cited by 1 | Viewed by 339
Abstract
In the author’s previous studies, new infinite numbers, their properties, and calculations were introduced. These infinite numbers quantify infinity and offer new possibilities for solving complicated problems in mathematics and applied sciences in which infinity appears. The current study presents additional properties and [...] Read more.
In the author’s previous studies, new infinite numbers, their properties, and calculations were introduced. These infinite numbers quantify infinity and offer new possibilities for solving complicated problems in mathematics and applied sciences in which infinity appears. The current study presents additional properties and topics regarding infinite numbers, as well as a comparison between infinite numbers. In this way, complex problems with inequalities involving series of numbers, in addition to limits of functions of x ∈ ℝ and improper integrals, can be addressed and solved easily. Furthermore, this study introduces rotational infinite numbers. These are not single numbers but sets of infinite numbers produced as the vectors of ordinary infinite numbers are rotated in the complex plane. Some properties of rotational infinite numbers and their calculations are presented. The rotational infinity unit, its inverse, and its opposite number, as well as the angular velocity of rotational infinite numbers, are defined and illustrated. Based on the above, the Riemann zeta function is equivalently written as the sum of three rotational infinite numbers, and it is further investigated and analyzed from another point of view. Furthermore, this study reveals and proves interesting formulas relating to the Riemann zeta function that can elegantly and simply calculate complicated ratios of infinite series of numbers. Finally, the above theoretical results were verified by a computational numerical simulation, which confirms the correctness of the analytical results. In summary, rotational infinite numbers can be used to easily analyze and solve problems that are difficult or impossible to solve using other methods. Full article
(This article belongs to the Special Issue Special Functions with Applications)
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