Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 20 December 2024 | Viewed by 10437

Special Issue Editors


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Guest Editor
Department of Mathematics Education, Andong National University, Andong 36729, Republic of Korea
Interests: number theory and combinatorics; noncommutative ring; p-adic analysis and special functions; module theory; matrix analysis; big data and data analysis

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Guest Editor
Department of Mathematics and Statistics, Universiti Putra Malaysia, Serdang, 43400 Selangor, Malaysia
Interests: integral transforms and special functions; generalized functions; generalized hypergeometric functions; distributions; ultra-distributions; topological semigroups, fractional integro-differential equations; fractals and fractional inequalities; fuzzy soft sets and applications in decision making
Special Issues, Collections and Topics in MDPI journals

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Guest Editor
Department of Mathematics, Vedant College of Engineering and Technology (Rajasthan Technical University), Bundi, Rajasthan State, India
Interests: generalized hypergeometric functions; applications in Ramanujan’s work and combinatorial identities integral transforms; statistical distributions; probability theory (including geometrical probability theory)

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to showcase a compilation of both original research articles and review articles that cover any aspect of Generalised Hypergeometric Functions, Integral Transforms and Number Theory and related study areas with a wide range of applications (in particular, applications in Ramanujan's work).

In order to encourage the ongoing efforts to produce effective transformations and applications related to the analysis of transformation, we invite mathematicians, researchers, and investigators to submit original research articles as well as review articles.

We look forward to receiving your contributions.

Dr. Dongkyu Lim
Prof. Dr. Adem Kilicman
Prof. Dr. Arjun Kumar Rathie
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. 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

  • generalized hypergeometric functions with one and more variables
  • transformation
  • summation
  • reduction formulas
  • applications in Ramanujan’s work and in combinatorics
  • integral transforms
  • Laplace
  • Mellin
  • Hankel transforms
  • Fourier transforms (including Sine and Cosine)
  • number theory with applications

Published Papers (11 papers)

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Research

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16 pages, 1494 KiB  
Article
Generalization of the Distance Fibonacci Sequences
by Nur Şeyma Yilmaz, Andrej Włoch and Engin Özkan
Axioms 2024, 13(7), 420; https://doi.org/10.3390/axioms13070420 - 21 Jun 2024
Viewed by 580
Abstract
In this study, we introduced a generalization of distance Fibonacci sequences and investigate some of its basic properties. We then proposed a generalization of distance Fibonacci sequences for negative integers and investigated some basic properties. Additionally, we explored the construction of matrix generators [...] Read more.
In this study, we introduced a generalization of distance Fibonacci sequences and investigate some of its basic properties. We then proposed a generalization of distance Fibonacci sequences for negative integers and investigated some basic properties. Additionally, we explored the construction of matrix generators for these sequences and offered a graphical representation to clarify their structure. Furthermore, we demonstrated how these generalizations can be applied to obtain the Padovan and Narayana sequences for specific parameter values. Full article
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18 pages, 290 KiB  
Article
The New G-Double-Laplace Transforms and One-Dimensional Coupled Sine-Gordon Equations
by Hassan Eltayeb and Said Mesloub
Axioms 2024, 13(6), 385; https://doi.org/10.3390/axioms13060385 - 5 Jun 2024
Viewed by 394
Abstract
This paper establishes a novel technique, which is called the G-double-Laplace transform. This technique is an extension of the generalized Laplace transform. We study its properties with examples and various theorems related to the G-double-Laplace transform that have been addressed and proven. Finally, [...] Read more.
This paper establishes a novel technique, which is called the G-double-Laplace transform. This technique is an extension of the generalized Laplace transform. We study its properties with examples and various theorems related to the G-double-Laplace transform that have been addressed and proven. Finally, we apply the G-double-Laplace transform decomposition method to solve the nonlinear sine-Gordon and coupled sine-Gordon equations. This method is a combination of the G-double-Laplace transform and decomposition method. In addition, some examples are examined to establish the accuracy and effectiveness of this technique. Full article
17 pages, 303 KiB  
Article
Generalized Limit Theorem for Mellin Transform of the Riemann Zeta-Function
by Antanas Laurinčikas and Darius Šiaučiūnas
Axioms 2024, 13(4), 251; https://doi.org/10.3390/axioms13040251 - 10 Apr 2024
Viewed by 782
Abstract
In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform Z(s), s=σ+it, with fixed [...] Read more.
In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform Z(s), s=σ+it, with fixed 1/2<σ<1, of the square |ζ(1/2+it)|2 of the Riemann zeta-function. We consider probability measures defined by means of Z(σ+iφ(t)), where φ(t), tt0>0, is an increasing to + differentiable function with monotonically decreasing derivative φ(t) satisfying a certain normalizing estimate related to the mean square of the function Z(σ+iφ(t)). This allows us to extend the distribution laws for Z(s). Full article
17 pages, 329 KiB  
Article
3F4 Hypergeometric Functions as a Sum of a Product of 2F3 Functions
by Jack C. Straton
Axioms 2024, 13(3), 203; https://doi.org/10.3390/axioms13030203 - 18 Mar 2024
Viewed by 1072
Abstract
This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3 functions expanded in sums of pair products of [...] Read more.
This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3 functions expanded in sums of pair products of 1F2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, 2F1 functions, and 3F2 functions into the realm of pFq functions where p<q for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. Full article
23 pages, 343 KiB  
Article
On Universality of Some Beurling Zeta-Functions
by Andrius Geštautas and Antanas Laurinčikas
Axioms 2024, 13(3), 145; https://doi.org/10.3390/axioms13030145 - 23 Feb 2024
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Abstract
Let P be the set of generalized prime numbers, and ζP(s), s=σ+it, denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using [...] Read more.
Let P be the set of generalized prime numbers, and ζP(s), s=σ+it, denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζP(s+iτ), τR. We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set {logp:pP}, and the existence of a bounded mean square for ζP(s). Under the above hypotheses, we obtain the universality of the function ζP(s). This means that the set of shifts ζP(s+iτ) approximating a given analytic function defined on a certain strip σ^<σ<1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied. Full article
0 pages, 381 KiB  
Article
The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving 1F2 Hypergeometric Functions That Arise from Them
by Jack C. Straton
Axioms 2024, 13(2), 134; https://doi.org/10.3390/axioms13020134 - 19 Feb 2024
Cited by 1 | Viewed by 1039
Abstract
The Bessel function of the first kind JNkx is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind INkx. The purpose of these expansions in Legendre polynomials was not an [...] Read more.
The Bessel function of the first kind JNkx is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind INkx. The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for JNkx useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving  1F2 hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1/2i3j5k7l11m13n17o19p multiplying powers of the coefficient k. Full article
23 pages, 354 KiB  
Article
Integral Representations over Finite Limits for Quantum Amplitudes
by Jack C. Straton
Axioms 2024, 13(2), 120; https://doi.org/10.3390/axioms13020120 - 14 Feb 2024
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Abstract
We extend previous research to derive three additional M-1-dimensional integral representations over the interval [0,1]. The prior version covered the interval [0,]. This extension applies to products of M Slater orbitals, since they [...] Read more.
We extend previous research to derive three additional M-1-dimensional integral representations over the interval [0,1]. The prior version covered the interval [0,]. This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) |x1x2|=x122x1x2cosθ+x22 to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce provide alternatives to Fourier transforms and are much more compact. The latter introduce a 3M-dimensional momentum integral for M products of Slater orbitals (in M separate denominators), followed in many cases by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current and prior methods are also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals while simultaneously moving them into a single (spatial) quadratic form in a common exponential. One may also use addition theorems for extracting the angular variables or even direct integration at times. Each method has its strengths and weaknesses. We found that these M-1-dimensional integral representations over the interval [0,1] are numerically stable, as was the prior version, having integrals running over the interval [0,], and one does not need to test for a sufficiently large upper integration limit as one does for the latter approach. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over [0,1] than over [0,]. In particular, the results of both prior and current representations have integration variables residing within square roots asarguments of Macdonald functions. In a number of cases, these can be converted to Meijer G-functions whose arguments have the form (ax2+bx+c)/x, for which a single tabled integral exists for the integrals from running over the interval [0,] of the prior paper, and from which other forms can be found using the techniques given therein. This is not so for integral representations over the interval [0,1]. Finally, we introduce a fourth integral representation that is not easily generalizable to large M but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over [0,1]. Full article
22 pages, 333 KiB  
Article
On the Multi-Dimensional Sumudu-Generalized Laplace Decomposition Method and Generalized Pseudo-Parabolic Equations
by Hassan Eltayeb
Axioms 2024, 13(2), 91; https://doi.org/10.3390/axioms13020091 - 30 Jan 2024
Viewed by 823
Abstract
The essential goal of this work is to suggest applying the multi-dimensional Sumdu generalized Laplace transform decomposition for solving pseudo-parabolic equations. This method is a combination of the multi-dimensional Sumudu transform, the generalized Laplace transform, and the decomposition method. We provided some examples [...] Read more.
The essential goal of this work is to suggest applying the multi-dimensional Sumdu generalized Laplace transform decomposition for solving pseudo-parabolic equations. This method is a combination of the multi-dimensional Sumudu transform, the generalized Laplace transform, and the decomposition method. We provided some examples to show the effectiveness and the ability of this approach to solve linear and nonlinear problems. The results show that the proposed method is reliable and easy for obtaining approximate solutions of FPDEs and is more precise if we compare it with existing methods. Full article
15 pages, 288 KiB  
Article
On Hybrid Hyper k-Pell, k-Pell–Lucas, and Modified k-Pell Numbers
by Elen Viviani Pereira Spreafico, Paula Catarino and Paulo Vasco
Axioms 2023, 12(11), 1047; https://doi.org/10.3390/axioms12111047 - 11 Nov 2023
Viewed by 1043
Abstract
Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper [...] Read more.
Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. Full article
17 pages, 643 KiB  
Article
On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives
by Fahad Alsidrani, Adem Kılıçman and Norazak Senu
Axioms 2023, 12(9), 901; https://doi.org/10.3390/axioms12090901 - 21 Sep 2023
Viewed by 1459
Abstract
This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, to obtain an approximate solution for the bounded [...] Read more.
This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, to obtain an approximate solution for the bounded space variable ν. The Laplace transformation is used in the time-fractional derivative operator to enhance the proposed numerical methods’ performance and accuracy and find an approximate solution to time-fractional Fornberg–Whitham equations. To confirm the accuracy of the proposed methods, we evaluate homogeneous time-fractional Fornberg–Whitham equations in terms of non-integer order and variable coefficients. The obtained results of the modified methods are shown through tables and graphs. Full article
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Review

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16 pages, 344 KiB  
Review
Monogenity and Power Integral Bases: Recent Developments
by István Gaál
Axioms 2024, 13(7), 429; https://doi.org/10.3390/axioms13070429 - 26 Jun 2024
Viewed by 731
Abstract
Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled [...] Read more.
Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled “Monogenity and Power Integral Bases”. We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given. Full article
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