Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
1.6
Journals
Axioms
Special Issues
Special Functions and Related Topics, 2nd Edition
share announcement

Special Functions and Related Topics, 2nd Edition

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: 30 November 2026 | Viewed by 12974

Special Issue Editors

Prof. Dr. Slobodan B. Tričković

E-Mail Website
Guest Editor
Department of Mathematics, University of Niš, Nis, Serbia
Interests: special functions; number theory; numerical analysis; q-calculus
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Miomir Stankovic

E-Mail Website
Guest Editor
Mathematical Institute of the Serbian Academy of Sciences and Arts, 11001 Belgrade, Serbia
Interests: fractional q-calculus; numerical analysis in q-calculus; deformed functions; special functions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Special functions, including trigonometric functions, have been studied and used for centuries. They present an old branch of mathematics to which many great mathematicians made significant contributions. These include, among others, Bernoulli and Euler numbers and polynomials; Euler's gamma and beta functions; the Digamma (Psi) function; the Pochhammer symbol; Gauss hypergeometric series; Riemann, Hurwitz, and Lerch zeta functions (together with Dirichlet and Mathieu series); Bessel and Struve differential equations; Bessel functions; Fourier–Bessel and Dini series of Bessel functions; Abel's, Jacobi's, and Weierstrass' work on elliptic functions; and the polynomials of Legendre, Jacobi, Laguerre, and Hermite.

The need to introduce most of these special functions was to solve specific problems, and they were developed when it became clear that the existing elementary functions were not satisfying enough to describe many unsolved problems in mathematics and physics. Thus, it was suitable or necessary to present new results as infinite series, integrals, or through solutions of differential equations. F.W. Bessel contributed to the theory of special functions by systematically investigating the functions already considered by Bernoulli, Euler, Lagrange, Fourier, and others, whose research areas were mechanics, astronomy, and heat conduction.

Currently, the family of Bessel functions counts many: Bessel functions of the first and second kind, modified Bessel functions of the first and second kind, Struve functions, modified Struve functions, Lommel functions, and others find their way into numerous applications. One topic in the theory of Bessel functions is the functional series of mathematical physics, having great importance in engineering and techniques. The Fourier–Bessel family of infinite series, consisting of Neumann, Kapteyn, Schlömilch, and Dini series, involving Bessel functions of the first kind or some other functions, belong to the hypergeometric representation.

These functions appear whenever natural phenomena are studied, in engineering problems, and while performing numerical simulations. They also crop up in statistics, financial models, and economic analysis. Newton and Leibniz leveraged them in the solution of differential equations. Special functions have been continuously developing ever since. Many new special functions and applications have been discovered in the past thirty years.

Prof. Dr. Slobodan B. Tričković
Prof. Dr. Miomir Stankovic
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 250 words) can be sent to the Editorial Office for assessment.

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. Axioms is an international peer-reviewed open access monthly 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 2400 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

  • gamma function
  • Riemann zeta and related functions
  • Hurwitz zeta function
  • Psi (Digamma) function
  • Pochammer symbol
  • Gauss hypergeometric function
  • Bessel and related functions
  • Bernoulli numbers and polynomials
  • harmonic numbers
  • Schlömilch series

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • Reprint: MDPI Books provides the opportunity to republish successful Special Issues in book format, both online and in print.

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

Published Papers (16 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

16 pages, 318 KB  
Article
Complete Monotonicity and Reduction Formulas for Certain Kampé de Fériet Functions
by Dmitrii Karp and Elena Prilepkina
Axioms 2026, 15(5), 360; https://doi.org/10.3390/axioms15050360 - 12 May 2026
Viewed by 276
Abstract
We extend the classical Euler-type integral representations for the Appell functions F1, F2, and F3, to the appropriate Kampé de Fériet functions by using integration against the Meijer–Nørlund G-function. In particular, these representations provide analytic continuation [...] Read more.
We extend the classical Euler-type integral representations for the Appell functions F1, F2, and F3, to the appropriate Kampé de Fériet functions by using integration against the Meijer–Nørlund G-function. In particular, these representations provide analytic continuation of the corresponding Kampé de Fériet functions. We further focus on the following two applications. First, we obtain sufficient conditions for complete monotonicity on the positive quadrant for three families of the Kampé de Fériet functions. These conditions can be expressed directly in terms of parameters and imply, among other things, joint log-convexity and related inequalities for partial derivatives of the Kampé de Fériet functions. Second, we show how known reduction and transformation formulas for the Appell and the generalized hypergeometric functions can be lifted to Kampé de Fériet functions by concatenating parameter arrays via the integral representations. This yields several reduction formulas, including extensions of some classical and new product identities. Further combining integration against the Meijer–Nørlund G-function with Slater’s double series transformation we obtain several exotic identities for infinite sums of the generalized hypergeometric functions. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
35 pages, 494 KB  
Article
Long Journey from Stevenson’s Formally Complex Hypergeometric Polynomials to Real-by-Definition Romanovski-Routh Polynomials
by Gregory Natanson
Axioms 2026, 15(5), 343; https://doi.org/10.3390/axioms15050343 - 6 May 2026
Viewed by 323
Abstract
The paper links Stevenson’s formally complex hypergeometric polynomials to the real-by-definition Romanovski-Routh (R-Routh) polynomials. The spectral problem for Stevenson’s second-order normal ordinary differential equation (NODE) was formulated in such a way that it could be re-used for the two SLPs associated with the [...] Read more.
The paper links Stevenson’s formally complex hypergeometric polynomials to the real-by-definition Romanovski-Routh (R-Routh) polynomials. The spectral problem for Stevenson’s second-order normal ordinary differential equation (NODE) was formulated in such a way that it could be re-used for the two SLPs associated with the ‘trigonometric Rosen-Morse’ (t-RM) potential on the finite interval and the implicit Milson potential on the line (both solvable by the R-Routh polynomials). Namely, the sought-for eigenfunction was required to represent the principal Frobenius solutions at both minus- and plus-infinity. We refer to these boundary conditions as the ‘dual-PFS’ problem. The exact solvability of the former SLP with the trigonometric Liouville potential was then proven by taking into account that the Romanovski-Routh polynomial of degree n must have exactly n real zeros as well as that the discrete energy spectrum in question had no upper bound. As the direct consequence of this proof, we then found that the mentioned d-PFS problem Stevenson’s NODE and therefore the second SLP associated with the Milson potential on the line were exactly solvable via the quasi-rational solutions (q-RSs) composed of the R-Routh polynomials with degree-dependent indexes. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
11 pages, 278 KB  
Article
Perturbed Dirichlet Series and the Difference Operator
by Nianlian Wang, Jay Mehta and Shigeru Kanemitsu
Axioms 2026, 15(4), 277; https://doi.org/10.3390/axioms15040277 - 10 Apr 2026
Viewed by 425
Abstract
The most well-known perturbed Dirichlet series is the Hurwitz zeta-function. Its analytic continuation via the binomial expansion has been studied extensively, beginning with Wilton’s work. In this paper, we shall provide, above all things, two striking instances of the binomial expansion. One is [...] Read more.
The most well-known perturbed Dirichlet series is the Hurwitz zeta-function. Its analytic continuation via the binomial expansion has been studied extensively, beginning with Wilton’s work. In this paper, we shall provide, above all things, two striking instances of the binomial expansion. One is elucidation of Mikolás an integral formula for the Hurwitz zeta-function valid in the critical strip to the effect that it is a manifestation of the picking-up principle of the values at the poles of the gamma function of the binomial expansion. The other is a new proof of Hasse’s formula by the binomial expansion. Also, we show the effectiveness of the difference operator in dealing with a series of the form ∑n=0∞(n+a1)−s1(n+a2)−s2(n+a3)−s3⋯,Resj>2,j=1,2,⋯ where 0<aj≤1 or aj∈H (in the upper half-plane). Furthermore, elucidation of the above results is made in the light of the Hardy–Hecke transform. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
16 pages, 703 KB  
Article
Bounds for the Second Hankel Determinant and Its Inverse in Specific Function Classes
by Trailokya Panigrahi, Teodor Bulboacă and Shiba Prasad Dhal
Axioms 2026, 15(2), 130; https://doi.org/10.3390/axioms15020130 - 11 Feb 2026
Viewed by 406
Abstract
This paper presents a newly defined subclass of analytic functions and explores several significant properties within the class, which use for their definitions the q-analogues of the derivative and the subordinations. Thus, we tried to connect different notions of the q-calculus [...] Read more.
This paper presents a newly defined subclass of analytic functions and explores several significant properties within the class, which use for their definitions the q-analogues of the derivative and the subordinations. Thus, we tried to connect different notions of the q-calculus with those of the Geometric Function Theory of one variable function. We identify the bounds of the initial coefficients and found upper bounds of the Fekete–Szegő functional for these classes. We investigate the relationship between the coefficients of an univalent function and those of its inverse by examining the difference between their second Hankel determinants. Furthermore, we analyze the behavior of the quantity module of the difference between the second Hankel determinant of a function and the same determinant for its inverse. To improve the obtained results by finding sharp estimations remains an interesting open question. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
►▼ Show Figures

Figure 1

37 pages, 2440 KB  
Article
Consciousness as 4-Manifold Painlevé V Dynamics: From Quantum Topology to Classical Gamma Oscillations
by Michel Planat
Axioms 2026, 15(2), 124; https://doi.org/10.3390/axioms15020124 - 6 Feb 2026
Cited by 2 | Viewed by 925
Abstract
We propose a novel mathematical framework for understanding consciousness as a dynamical phenomenon governed by nonlinear integrable equations. The central hypothesis identifies conscious state dynamics with the Painlevé VI equation and its confluence limits, providing a unified description of stability, bifurcation, and collapse [...] Read more.
We propose a novel mathematical framework for understanding consciousness as a dynamical phenomenon governed by nonlinear integrable equations. The central hypothesis identifies conscious state dynamics with the Painlevé VI equation and its confluence limits, providing a unified description of stability, bifurcation, and collapse across cognitive regimes. In this approach, consciousness is modeled as an emergent phase sustained near criticality, where coherent quantum-like structures and classical decoherence coexist in a regulated balance. The theory is formulated in terms of isomonodromic deformations on SL(2,C) character varieties, allowing conscious states to be characterized by monodromy data and their controlled evolution. This geometric setting naturally encodes memory, attention, and transitions between conscious and unconscious phases, while confluence processes account for irreversible loss of coherence. A two-stage quantum-to-classical transition is identified, separating microscopic coherence from macroscopic stabilization. The framework yields universal signatures such as critical slowing down, scaling laws near transition points, and robustness under perturbations, linking consciousness dynamics to broader classes of critical phenomena observed in physics and complex systems. By replacing heuristic assumptions with a mathematically constrained dynamical structure, this work extends existing quantum consciousness models and provides a tractable platform for comparison with neural, biological, and informational data. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
►▼ Show Figures

Figure 1

18 pages, 2195 KB  
Article
On the Expansion of Legendre Polynomials in Bicomplex Space and Coupling with Fractional Operators
by Ahmed Bakhet, Shahid Hussain, Mohra Zayed and Aya M. Mourad
Axioms 2026, 15(1), 61; https://doi.org/10.3390/axioms15010061 - 15 Jan 2026
Viewed by 411
Abstract
In this paper, we introduce a novel version of the Legendre polynomials in the bicomplex system. We investigate the essential properties of the Legendre polynomial, focusing on its bicomplex structure, generating functions, orthogonality, and recurrence relations. We present a solution to the Legendre [...] Read more.
In this paper, we introduce a novel version of the Legendre polynomials in the bicomplex system. We investigate the essential properties of the Legendre polynomial, focusing on its bicomplex structure, generating functions, orthogonality, and recurrence relations. We present a solution to the Legendre differential equation in bicomplex space. Additionally, we discuss both theoretical and practical contributions, especially in bicomplex Riemann Liouville fractional calculus. We numerically study the construction of bicomplex Legendre polynomials, orthogonality, spectral projection, coefficient decay, and spectral convergence in bicomplex space. The findings contribute to a deeper insight into bicomplex functions, paving the way for further developments in science and mathematical analysis, and providing a foundation for future research on special functions and fractional operators within the bicomplex setting. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
►▼ Show Figures

Figure 1

20 pages, 391 KB  
Article
Integral Transforms in Number Theory
by Guodong Liu, Takako Kuzumaki and Shigeru Kanemitsu
Axioms 2025, 14(12), 917; https://doi.org/10.3390/axioms14120917 - 12 Dec 2025
Cited by 1 | Viewed by 844
Abstract
Integral transforms play a fundamental role in science and engineering. Above all, the Fourier transform is the most vital, which has some specifications—Laplace transform, Mellin transform, etc., with their inverse transforms. In this paper, we restrict ourselves to the use of a few [...] Read more.
Integral transforms play a fundamental role in science and engineering. Above all, the Fourier transform is the most vital, which has some specifications—Laplace transform, Mellin transform, etc., with their inverse transforms. In this paper, we restrict ourselves to the use of a few versions of the Mellin transform, which are best suited to the treatment of zeta functions as Dirichlet series. In particular, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta functions by considering some generalizations of the holomorphic and non-holomorphic Eisenstein series as the Epstein-type Eisenstein series, which have been treated as totally foreign subjects to each other. We restrict to the modular relations with one gamma factor and the resulting integrals reduce to a form of the modified Bessel function. In the H-function hierarchy, what we work with is the second simplest H1,11,1↔H0,22,0, with H denoting the Fox H-function. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
18 pages, 1074 KB  
Article
Third-Order Functional Differential Equations with Damping Term: Oscillatory Behavior of Solutions
by Asma Al-Jaser, Eman Alluqmani, Belgees Qaraad and Higinio Ramos
Axioms 2025, 14(12), 877; https://doi.org/10.3390/axioms14120877 - 28 Nov 2025
Viewed by 729
Abstract
In this paper, we investigate the asymptotic and oscillatory behavior of a specific class of third-order functional differential equations with damping terms and deviating arguments. By employing the comparison principle, Riccati transformation, and the integral averaging technique, we derive new criteria that guarantee [...] Read more.
In this paper, we investigate the asymptotic and oscillatory behavior of a specific class of third-order functional differential equations with damping terms and deviating arguments. By employing the comparison principle, Riccati transformation, and the integral averaging technique, we derive new criteria that guarantee all solutions to the studied equation oscillate when ∫⊤0∞1/γ1/α⊤d⊤=∞ and ϱ⊤≤ϱ0<∞. This study introduces novel conditions and effective analytical tools, which enhance our understanding of such equations and broaden their range of applications. Illustrative examples are provided to demonstrate the applicability of the results. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
►▼ Show Figures

Figure 1

21 pages, 395 KB  
Article
An Efficient Iteration Method for Fixed-Point Approximation and Its Application to Fractional Volterra–Fredholm Integro–Differential Equations
by Ekta Sharma, Shubham Kumar Mittal, Sunil Panday and Lorentz Jäntschi
Axioms 2025, 14(11), 830; https://doi.org/10.3390/axioms14110830 - 11 Nov 2025
Viewed by 1193
Abstract
This paper proposes an efficient iteration method for fixed-point approximation in Banach spaces. The method accelerates convergence by incorporating a squared operator term within the iteration process. Analytical proofs verify its convergence and stability. Comparative numerical tests show that it converges faster and [...] Read more.
This paper proposes an efficient iteration method for fixed-point approximation in Banach spaces. The method accelerates convergence by incorporating a squared operator term within the iteration process. Analytical proofs verify its convergence and stability. Comparative numerical tests show that it converges faster and more reliably than established Picard-type methods. Its application to fractional models involving the Gamma function highlights the method’s efficiency and potential for broader use in nonlinear and fractional systems. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
►▼ Show Figures

Figure 1

28 pages, 12538 KB  
Article
Embedding Vacuum Fluctuations in the Dirac Equation: On the Neutrino Electric Millicharge and Magnetic Moment
by Hector Eduardo Roman
Axioms 2025, 14(11), 779; https://doi.org/10.3390/axioms14110779 - 23 Oct 2025
Viewed by 947
Abstract
An extension of the Dirac equation for an initially massless particle carrying an electric charge, assumed to be embedded via minimal coupling into an external fluctuating electromagnetic four-potential of the vacuum, is suggested. We conjecture that appropriate averages of the four-vector can lead [...] Read more.
An extension of the Dirac equation for an initially massless particle carrying an electric charge, assumed to be embedded via minimal coupling into an external fluctuating electromagnetic four-potential of the vacuum, is suggested. We conjecture that appropriate averages of the four-vector can lead to observable quantities, such as a particle mass in its rest frame. The conditions on the potential mean values to become gauge-invariant are obtained. The mass is found to be proportional to the magnitude of the charge times the associated mean Lorentz scalar of the four-potential, and the relation holds for both spacelike and timelike types of four-vectors. For the latter, the extended Dirac equation violates Lorentz covariance, but the violation can be argued to occur within a time scale allowed by the uncertainty principle. For larger times, the particle has acquired a mass and Lorentz covariance is restored. This mathematical scenario is applied to acquire estimates of the neutrino millicharge and magnetic moment, in good agreement with the present upper bounds obtained experimentally. The issue of unstable particle decay is considered by focusing, for illustration, on the main decay channels of the selected particles. From the lifetime of the τ lepton, a lower bound of the effective neutrino mass is predicted, which can be tested in future experiments. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
►▼ Show Figures

Graphical abstract

29 pages, 388 KB  
Article
Harmonic Series of Convergence Ratio “1/4" with Cubic Central Binomial Coefficients
by Chunli Li and Wenchang Chu
Axioms 2025, 14(11), 776; https://doi.org/10.3390/axioms14110776 - 23 Oct 2025
Viewed by 732
Abstract
We examine a useful hypergeometric transformation formula by means of the coefficient extraction method. A large class of “binomial/harmonic series” (of convergence ratio “1/4”) containing the cubic central binomial coefficients and harmonic numbers is systematically investigated. Numerous closed summation [...] Read more.
We examine a useful hypergeometric transformation formula by means of the coefficient extraction method. A large class of “binomial/harmonic series” (of convergence ratio “1/4”) containing the cubic central binomial coefficients and harmonic numbers is systematically investigated. Numerous closed summation formulae are established, including a remarkable series about harmonic numbers of the third order. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
12 pages, 282 KB  
Article
The Cotangent Function as an Avatar of the Polylogarithm Function of Order 0 and Ramanujan’s Formula
by Ruiyang Li, Haoyang Lu and Shigeru Kanemitsu
Axioms 2025, 14(10), 774; https://doi.org/10.3390/axioms14100774 - 21 Oct 2025
Cited by 1 | Viewed by 746
Abstract
In this paper we will be concerned with zeta-symmetry—the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)—and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason [...] Read more.
In this paper we will be concerned with zeta-symmetry—the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)—and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason is that the cotangent function (as a function in the upper half-plane, say) is the polylogarithm function of order 0 (with complex exponential argument), and therefore it shares properties intrinsic to the Lerch zeta-function of order 0. Here we view the Lerch zeta-function defined in the unit circle as a zeta-function in a wider sense, as a function defined in the upper and lower half-planes. As evidence, we give a plausibly most natural proof of Ramanujan’s formula, including the eta transformation formula as a consequence of the modular relation via the cotangent function, speculating the reason why Ramanujan had been led to such a formula. Other evidence includes the pre-Poisson summation formula as the pick-up principle (which in turn is a generalization of the argument principle). Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
24 pages, 1300 KB  
Article
On the Construction and Analysis of a Fractional-Order Dirac Delta Distribution with Application
by Muhammad Muddassar, Adil Jhangeer, Nasir Siddiqui, Malik Sajjad Mehmood, Liaqat Khan and Tahira Jabeen
Axioms 2025, 14(10), 728; https://doi.org/10.3390/axioms14100728 - 26 Sep 2025
Viewed by 1052
Abstract
We introduce the generalized fractional-order Dirac delta distribution δGFODDF, defined by applying the generalized fractional derivative (GFD) operator to the Heaviside function. This construction extends the classical Dirac delta to non-integer orders, allowing modeling of systems with memory and non-local effects. [...] Read more.
We introduce the generalized fractional-order Dirac delta distribution δGFODDF, defined by applying the generalized fractional derivative (GFD) operator to the Heaviside function. This construction extends the classical Dirac delta to non-integer orders, allowing modeling of systems with memory and non-local effects. We establish fundamental properties—including shifting, scaling, evenness, derivative, and convolution—within a rigorous distributional framework and present explicit proofs. Applications are demonstrated by solving linear fractional differential equations and by modeling drug release with fractional kinetics, where the new delta captures impulse responses with long-term memory. Numerical illustrations confirm that δGFODDF reduces to the classical delta when η=1, while providing additional flexibility for 0<η<1. These results show that δGFODDF is a powerful tool for fractional-order analysis in mathematics, physics, and biomedical engineering. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
►▼ Show Figures

Figure 1

16 pages, 3147 KB  
Article
A Note on Multi-Index Mittag-Leffler Functions and Parametric Laguerre-Type Exponentials
by Hari Mohan Srivastava, Diego Caratelli and Paolo Emilio Ricci
Axioms 2025, 14(9), 705; https://doi.org/10.3390/axioms14090705 - 18 Sep 2025
Cited by 2 | Viewed by 1009
Abstract
This paper explores the eigenfunctions of specific Laguerre-type parametric operators to develop multi-parametric models, which are associated with a class of the generalized Mittag-Leffler type functions, for dynamical systems and population dynamics. By leveraging these multi-parametric approaches, we introduce new concepts in number [...] Read more.
This paper explores the eigenfunctions of specific Laguerre-type parametric operators to develop multi-parametric models, which are associated with a class of the generalized Mittag-Leffler type functions, for dynamical systems and population dynamics. By leveraging these multi-parametric approaches, we introduce new concepts in number theory, specifically those involving multi-parametric Bernoulli and Euler numbers, along with other related polynomials. Several numerical examples, which are generated by using the computer algebra program Mathematica© (Version 14.3), demonstrate the effectiveness of the models that we have presented and analyzed in this paper. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
►▼ Show Figures

Figure 1

18 pages, 316 KB  
Article
On the Laplace-Type Transform and Its Applications
by Slobodan B. Tričković and Miomir S. Stanković
Axioms 2025, 14(9), 697; https://doi.org/10.3390/axioms14090697 - 15 Sep 2025
Viewed by 1096
Abstract
Using the Laplace transform and the Gamma function, we obtain the Laplace-type transform, with the property of mapping a function to a functional sequence, which cannot be realized by the Laplace transform. In addition, we construct a backward difference as a generalization of [...] Read more.
Using the Laplace transform and the Gamma function, we obtain the Laplace-type transform, with the property of mapping a function to a functional sequence, which cannot be realized by the Laplace transform. In addition, we construct a backward difference as a generalization of the backward difference operator ∇. By connecting it to the Laplace-type transform, we deduce a method for solving difference equations and, relying on classical orthogonal polynomials, for obtaining combinatorial identities. A table of some elementary functions and their images is at the end of the text. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)

Review

Jump to: Research

28 pages, 13837 KB  
Review
Spacetime Metrics with Spherical Symmetry: A Short Review on the Riemann Tensors and Kretschmann Scalars
by Hector Eduardo Roman
Axioms 2026, 15(4), 264; https://doi.org/10.3390/axioms15040264 - 5 Apr 2026
Viewed by 559
Abstract
While the standard Schwarzschild metric is overwhelmingly employed in general relativity (GR) as the starting point for various spherical spacetime metric calculations, its isotropic (ISO) form is mentioned in more specialized contexts and its derivation is barely discussed in published GR literature. In [...] Read more.
While the standard Schwarzschild metric is overwhelmingly employed in general relativity (GR) as the starting point for various spherical spacetime metric calculations, its isotropic (ISO) form is mentioned in more specialized contexts and its derivation is barely discussed in published GR literature. In this work, we review the isotropic metric, stressing that it stands out as a useful spherically symmetric metric to be employed also in traditional GR problems. We start by deriving the ISO metric through solving the vacuum field equations in Cartesian coordinates, thereby obtaining the Ricci tensor also in spherical coordinates. We then analytically calculate the Riemann tensor in Cartesian coordinates, proving its consistency with the Ricci tensor calculation for pedagogical reasons. Finally, from the Riemann tensor we exactly evaluate the Kretschmann scalar, which lacks metric singularities, a result consistent with the known singular behavior of the standard Schwarzschild metric. We conclude that the isotropic metric naturally emerges as a suitable candidate for modeling static neutron stars and regular black holes, thereby complementing the present attempts to understand these rapidly evolving research fields. Full article
(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)
►▼ Show Figures

Figure 1

Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-16
Back to TopTop