Higher Transcendental Functions and Their Multi-Disciplinary Applications

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

Deadline for manuscript submissions: closed (15 August 2022) | Viewed by 29222

Special Issue Editor


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Guest Editor
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: real and complex analysis; fractional calculus and its applications; integral equations and transforms; higher transcendental functions and their applications; q-series and q-polynomials; analytic number theory; analytic and geometric Inequalities; probability and statistics; inventory modeling and optimization
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Special Issue Information

Dear Colleagues,

This Special Issue cordially invites and welcomes review, expository, and original research articles dealing with the recent advances on various potentially useful families of special functions (or, more precisely, higher transcendental functions) of mathematical analysis, mathematical physics, analytic number theory, and the geometric function theory of complex analysis, as well as their applications in many widely-scattered disciplines within the physical, biological, chemical, earth, engineering and statistical sciences.

We look forward to your contributions to this Special Issue.

Prof. Dr. Hari Mohan Srivastava
Guest Editor

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Keywords

  • Mathematical (or higher transcendental) functions and their applications
  • Fractional-order derivatives and integrals and their applications
  • q-differences (or q-derivatives), q-series and q-polynomials
  • Functions of analytic number theory
  • Special functions of mathematical physics and applied mathematics
  • Geometric function theory of complex analysis

Published Papers (18 papers)

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Editorial

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3 pages, 162 KiB  
Editorial
Higher Transcendental Functions and Their Multi-Disciplinary Applications
by Hari Mohan Srivastava
Mathematics 2022, 10(24), 4740; https://doi.org/10.3390/math10244740 - 14 Dec 2022
Viewed by 1350
Abstract
This volume consists of a collection of 17 peer-reviewed and accepted submissions from authors around the world (including several invited feature articles) to the Special Issue of the journal Mathematics, on the general subject-area of “Higher Transcendental Functions and Their Multi-Disciplinary Applications” [...] Read more.
This volume consists of a collection of 17 peer-reviewed and accepted submissions from authors around the world (including several invited feature articles) to the Special Issue of the journal Mathematics, on the general subject-area of “Higher Transcendental Functions and Their Multi-Disciplinary Applications” [...] Full article

Research

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15 pages, 318 KiB  
Article
The Exact Solutions for Several Partial Differential-Difference Equations with Constant Coefficients
by Hongyan Xu, Ling Xu and Hari Mohan Srivastava
Mathematics 2022, 10(19), 3596; https://doi.org/10.3390/math10193596 - 1 Oct 2022
Cited by 2 | Viewed by 1097
Abstract
This article is concerned with the description of the entire solutions of several Fermat type partial differential-difference equations (PDDEs) [...] Read more.
This article is concerned with the description of the entire solutions of several Fermat type partial differential-difference equations (PDDEs) μf(z)+λfz1(z)2+[αf(z+c)βf(z)]2=1, and μf(z)+λ1fz1(z)+λ2fz2(z)2+[αf(z+c)βf(z)]2=1, where fz1(z)=fz1 and fz2(z)=fz2, c=(c1,c2)C2, α,β,μ,λ,λ1,λ2,c1,c2 are constants in C. Our theorems in this paper give some descriptions of the forms of transcendental entire solutions for the above equations, which are some extensions and improvement of the previous theorems given by Xu, Cao, Liu, and Yang. In particular, we exhibit a series of examples to explain that the existence conditions and the forms of transcendental entire solutions with a finite order of such equations are precise. Full article
11 pages, 279 KiB  
Article
Characterizations of Continuous Fractional Bessel Wavelet Transforms
by Hari M. Srivastava, Kush Kumar Mishra and Santosh K. Upadhyay
Mathematics 2022, 10(17), 3084; https://doi.org/10.3390/math10173084 - 27 Aug 2022
Cited by 10 | Viewed by 1135
Abstract
In this paper, we present a systematic study of the various characteristics and properties of some continuous and discrete fractional Bessel wavelet transforms. The method is based upon the theory of the fractional Hankel transform. Full article
19 pages, 345 KiB  
Article
Improper Integrals Involving Powers of Inverse Trigonometric and Hyperbolic Functions
by Chunli Li and Wenchang Chu
Mathematics 2022, 10(16), 2980; https://doi.org/10.3390/math10162980 - 18 Aug 2022
Cited by 5 | Viewed by 1380
Abstract
Three classes of improper integrals involving higher powers of arctanh, arctan, and arcsin are examined using the recursive approach. Numerous explicit formulae are established, which evaluate these integrals in terms of π, ln2, the Riemann zeta function, and the [...] Read more.
Three classes of improper integrals involving higher powers of arctanh, arctan, and arcsin are examined using the recursive approach. Numerous explicit formulae are established, which evaluate these integrals in terms of π, ln2, the Riemann zeta function, and the Dirichlet beta function. Full article
15 pages, 342 KiB  
Article
Some Generalized Properties of Poly-Daehee Numbers and Polynomials Based on Apostol–Genocchi Polynomials
by Talha Usman, Nabiullah Khan, Mohd Aman, Shrideh Al-Omari, Kamsing Nonlaopon and Junesang Choi
Mathematics 2022, 10(14), 2502; https://doi.org/10.3390/math10142502 - 18 Jul 2022
Cited by 4 | Viewed by 1254
Abstract
Numerous polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields. The purpose of this research is to establish a unified class of Apostol–Genocchi polynomials based on poly-Daehee polynomials and to explore some of their [...] Read more.
Numerous polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields. The purpose of this research is to establish a unified class of Apostol–Genocchi polynomials based on poly-Daehee polynomials and to explore some of their features and identities. We investigate these polynomials via generating functions and deduce various identities, summation formulae, differential and integral formulas, implicit summation formulae, and several characterized generating functions for new numbers and polynomials. Finally, by using an operational version of Apostol–Genocchi polynomials, we derive some results in terms of new special polynomials. Due to the generic nature of the findings described here, they are used to reduce and generate certain known or novel formulae and identities for relatively simple polynomials and numbers. Full article
14 pages, 303 KiB  
Article
Bézier-Summation-Integral-Type Operators That Include Pólya–Eggenberger Distribution
by Syed Abdul Mohiuddine, Arun Kajla and Abdullah Alotaibi
Mathematics 2022, 10(13), 2222; https://doi.org/10.3390/math10132222 - 25 Jun 2022
Cited by 6 | Viewed by 1297
Abstract
We define the summation-integral-type operators involving the ideas of Pólya–Eggenberger distribution and Bézier basis functions, and study some of their basic approximation properties. In addition, by means of the Ditzian–Totik modulus of smoothness, we study a direct theorem as well as a quantitative [...] Read more.
We define the summation-integral-type operators involving the ideas of Pólya–Eggenberger distribution and Bézier basis functions, and study some of their basic approximation properties. In addition, by means of the Ditzian–Totik modulus of smoothness, we study a direct theorem as well as a quantitative Voronovskaja-type theorem for our newly constructed operators. Moreover, we investigate the approximation of functions with derivatives of bounded variation (DBV) of the aforesaid operators. Full article
26 pages, 410 KiB  
Article
A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices
by Hongyan Xu, Guangsheng Chen, Hari Mohan Srivastava, Hong Li, Zuxing Xuan and Yongqin Cui
Mathematics 2022, 10(13), 2220; https://doi.org/10.3390/math10132220 - 24 Jun 2022
Cited by 1 | Viewed by 1148
Abstract
In this paper, our first purpose is to describe a class of phenomena involving the growth in the Hadamard–Kong product of several Dirichlet series with different growth indices. We prove that (i) the order of the Hadamard–Kong product series is determined by the [...] Read more.
In this paper, our first purpose is to describe a class of phenomena involving the growth in the Hadamard–Kong product of several Dirichlet series with different growth indices. We prove that (i) the order of the Hadamard–Kong product series is determined by the growth in the Dirichlet series with smaller indices if these Dirichlet series have different growth indices; (ii) the q1-type of the Hadamard–Kong product series is equal to zero if p Dirichlet series are of qj-regular growth, and q1<q2<<qp, qjN+, j=1,2,,p. The second purpose is to reveal the properties of the growth in the Hadamard–Kong product series of two Dirichlet series—when one Dirichlet series is of finite order, the other is of logarithmic order, and two Dirichlet series are of finite logarithmic order—and obtain the growth relationships between the Hadamard–Kong product series and two Dirchlet series concerning the order, the logarithmic order, logarithmic type, etc. Finally, some examples are given to show that our results are best possible. Full article
14 pages, 311 KiB  
Article
Stancu-Type Generalized q-Bernstein–Kantorovich Operators Involving Bézier Bases
by Wen-Tao Cheng, Md Nasiruzzaman and Syed Abdul Mohiuddine
Mathematics 2022, 10(12), 2057; https://doi.org/10.3390/math10122057 - 14 Jun 2022
Cited by 6 | Viewed by 1199
Abstract
We construct the Stancu-type generalization of q-Bernstein operators involving the idea of Bézier bases depending on the shape parameter 1ζ1 and obtain auxiliary lemmas. We discuss the local approximation results in term of a Lipschitz-type function based [...] Read more.
We construct the Stancu-type generalization of q-Bernstein operators involving the idea of Bézier bases depending on the shape parameter 1ζ1 and obtain auxiliary lemmas. We discuss the local approximation results in term of a Lipschitz-type function based on two parameters and a Lipschitz-type maximal function, as well as other related results for our newly constructed operators. Moreover, we determine the rate of convergence of our operators by means of Peetre’s K-functional and corresponding modulus of continuity. Full article
21 pages, 330 KiB  
Article
Approximation of GBS Type q-Jakimovski-Leviatan-Beta Integral Operators in Bögel Space
by Abdullah Alotaibi
Mathematics 2022, 10(5), 675; https://doi.org/10.3390/math10050675 - 22 Feb 2022
Cited by 4 | Viewed by 1310
Abstract
In the present article, we introduce the bivariate variant of Beta integral type operators based on Appell polynomials via q-calculus. We study the local and global type approximation properties for these new operators. Next, we introduce the GBS form for these new [...] Read more.
In the present article, we introduce the bivariate variant of Beta integral type operators based on Appell polynomials via q-calculus. We study the local and global type approximation properties for these new operators. Next, we introduce the GBS form for these new operators and then study the degree of approximation by means of modulus of smoothness, mixed modulus of smoothness and Lipschitz class of Bögel continuous functions. Full article
17 pages, 410 KiB  
Article
Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media
by Alexander V. Shapovalov and Anton E. Kulagin
Mathematics 2021, 9(23), 2995; https://doi.org/10.3390/math9232995 - 23 Nov 2021
Cited by 6 | Viewed by 1100
Abstract
A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical [...] Read more.
A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given. Full article
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10 pages, 264 KiB  
Article
Analytical Investigation of the Existence of Solutions for a System of Nonlinear Hadamard-Type Integro-Differential Equations Based upon the Multivariate Mittag-Leffler Function
by Chenkuan Li, Rekha Srivastava and Kyle Gardiner
Mathematics 2021, 9(21), 2733; https://doi.org/10.3390/math9212733 - 28 Oct 2021
Cited by 1 | Viewed by 1307
Abstract
In this paper, the authors propose an investigation of the existence of solutions for a system of nonlinear Hadamard-type integro-differential equations in a Banach space. The result derived is new and based upon Babenko’s approach, Leray-Schauder’s nonlinear alternative, and the multivariate Mittag-Leffler function. [...] Read more.
In this paper, the authors propose an investigation of the existence of solutions for a system of nonlinear Hadamard-type integro-differential equations in a Banach space. The result derived is new and based upon Babenko’s approach, Leray-Schauder’s nonlinear alternative, and the multivariate Mittag-Leffler function. Using an illustrative example, a demonstration of the application of the main theorem is also considered. Full article
20 pages, 868 KiB  
Article
Multi-Step Inertial Regularized Methods for Hierarchical Variational Inequality Problems Involving Generalized Lipschitzian Mappings
by Bingnan Jiang, Yuanheng Wang and Jen-Chih Yao
Mathematics 2021, 9(17), 2103; https://doi.org/10.3390/math9172103 - 31 Aug 2021
Cited by 7 | Viewed by 1569
Abstract
In this paper, we construct two multi-step inertial regularized methods for hierarchical inequality problems involving generalized Lipschitzian and hemicontinuous mappings in Hilbert spaces. Then we present two strong convergence theorems and some numerical experiments to show the effectiveness and feasibility of our new [...] Read more.
In this paper, we construct two multi-step inertial regularized methods for hierarchical inequality problems involving generalized Lipschitzian and hemicontinuous mappings in Hilbert spaces. Then we present two strong convergence theorems and some numerical experiments to show the effectiveness and feasibility of our new iterative methods. Full article
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15 pages, 9978 KiB  
Article
A Link between Approximation Theory and Summability Methods via Four-Dimensional Infinite Matrices
by Hari M. Srivastava, Khursheed J. Ansari, Faruk Özger and Zeynep Ödemiş Özger
Mathematics 2021, 9(16), 1895; https://doi.org/10.3390/math9161895 - 9 Aug 2021
Cited by 35 | Viewed by 2852
Abstract
In this study, we present a link between approximation theory and summability methods by constructing bivariate Bernstein-Kantorovich type operators on an extended domain with reparametrized knots. We use a statistical convergence type and power series method to obtain certain Korovkin type theorems, and [...] Read more.
In this study, we present a link between approximation theory and summability methods by constructing bivariate Bernstein-Kantorovich type operators on an extended domain with reparametrized knots. We use a statistical convergence type and power series method to obtain certain Korovkin type theorems, and we study certain rates of convergences related to these summability methods. Furthermore, we numerically analyze the theoretical results and provide some computer graphics to emphasize the importance of this study. Full article
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14 pages, 315 KiB  
Article
A Class of k-Symmetric Harmonic Functions Involving a Certain q-Derivative Operator
by Hari M. Srivastava, Nazar Khan, Shahid Khan, Qazi Zahoor Ahmad and Bilal Khan
Mathematics 2021, 9(15), 1812; https://doi.org/10.3390/math9151812 - 30 Jul 2021
Cited by 18 | Viewed by 1641
Abstract
In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, [...] Read more.
In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized q-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called (p,q)-variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary. Full article
15 pages, 822 KiB  
Article
Gottlieb Polynomials and Their q-Extensions
by Esra ErkuŞ-Duman and Junesang Choi
Mathematics 2021, 9(13), 1499; https://doi.org/10.3390/math9131499 - 26 Jun 2021
Cited by 2 | Viewed by 1386
Abstract
Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating [...] Read more.
Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating functions for three sequences associated with a finite power series whose coefficients are products of the known q-extended multivariable and multiparameter Gottlieb polynomials and another non-vanishing multivariable function. Furthermore, numerous possible particular cases of our main identities are considered. Finally, we return to Khan and Asif’s q-Gottlieb polynomials to highlight certain connections with several other known q-polynomials, and provide its q-integral representation. Furthermore, we conclude this paper by disclosing our future investigation plan. Full article
18 pages, 326 KiB  
Article
Special Functions as Solutions to the Euler–Poisson–Darboux Equation with a Fractional Power of the Bessel Operator
by Azamat Dzarakhohov, Yuri Luchko and Elina Shishkina
Mathematics 2021, 9(13), 1484; https://doi.org/10.3390/math9131484 - 24 Jun 2021
Cited by 6 | Viewed by 1809
Abstract
In this paper, we consider fractional ordinary differential equations and the fractional Euler–Poisson–Darboux equation with fractional derivatives in the form of a power of the Bessel differential operator. Using the technique of the Meijer integral transform and its modification, fundamental solutions to these [...] Read more.
In this paper, we consider fractional ordinary differential equations and the fractional Euler–Poisson–Darboux equation with fractional derivatives in the form of a power of the Bessel differential operator. Using the technique of the Meijer integral transform and its modification, fundamental solutions to these equations are derived in terms of the Fox–Wright function, the Fox H-function, and their particular cases. We also provide some explicit formulas for the solutions to the corresponding initial-value problems in terms of the generalized convolutions introduced in this paper. Full article
12 pages, 264 KiB  
Article
A Study of Some Families of Multivalent q-Starlike Functions Involving Higher-Order q-Derivatives
by Bilal Khan, Zhi-Guo Liu, Hari M. Srivastava, Nazar Khan, Maslina Darus and Muhammad Tahir
Mathematics 2020, 8(9), 1470; https://doi.org/10.3390/math8091470 - 1 Sep 2020
Cited by 45 | Viewed by 2047
Abstract
In the present investigation, by using certain higher-order q-derivatives, the authors introduce and investigate several new subclasses of the family of multivalent q-starlike functions in the open unit disk. For each of these newly-defined function classes, several interesting properties and characteristics [...] Read more.
In the present investigation, by using certain higher-order q-derivatives, the authors introduce and investigate several new subclasses of the family of multivalent q-starlike functions in the open unit disk. For each of these newly-defined function classes, several interesting properties and characteristics are systematically derived. These properties and characteristics include (for example) distortion theorems and radius problems. A number of coefficient inequalities and a sufficient condition for functions belonging to the subclasses studied here are also discussed. Relevant connections of the various results presented in this investigation with those in earlier works on this subject are also pointed out. Full article

Review

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23 pages, 402 KiB  
Review
Some Families of Generating Functions Associated with Orthogonal Polynomials and Other Higher Transcendental Functions
by Hari Mohan Srivastava
Mathematics 2022, 10(20), 3730; https://doi.org/10.3390/math10203730 - 11 Oct 2022
Cited by 15 | Viewed by 1518
Abstract
In this invited survey-cum-expository review article, we present a brief and comprehensive account of some general families of linear and bilinear generating functions which are associated with orthogonal polynomials and such other higher transcendental functions as (for example) hypergeometric functions and hypergeometric polynomials [...] Read more.
In this invited survey-cum-expository review article, we present a brief and comprehensive account of some general families of linear and bilinear generating functions which are associated with orthogonal polynomials and such other higher transcendental functions as (for example) hypergeometric functions and hypergeometric polynomials in one, two and more variables. Many of the results as well as the methods and techniques used for their derivations, which are presented here, are intended to provide incentive and motivation for further research on the subject investigated in this article. Full article
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