Editorial

3 pages, 162 KiB

Open AccessEditorial

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 1227

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

Research

Jump to: Editorial, Review

15 pages, 318 KiB

Open AccessArticle

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 - 01 Oct 2022

Cited by 2 | Viewed by 1004

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)

{[μ f (z) + λ f_{z_{1}} (z)]}^{2} + {[α f (z + c) - β f (z)]}^{2} = 1,

and

{[μ f (z) + λ_{1} f_{z_{1}} (z) + λ_{2} f_{z_{2}} (z)]}^{2} + {[α f (z + c) - β f (z)]}^{2} = 1,

where

f_{z_{1}} (z) = \frac{\partial f}{\partial z_{1}}

and

f_{z_{2}} (z) = \frac{\partial f}{\partial z_{2}}

,

c = (c_{1}, c_{2}) \in C^{2}

,

α, β, μ, λ, λ_{1}, λ_{2}, c_{1}, c_{2}

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

11 pages, 279 KiB

Open AccessArticle

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 1010

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

19 pages, 345 KiB

Open AccessFeature PaperArticle

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 4 | Viewed by 1241

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

,

ln 2

, 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

,

ln 2

, the Riemann zeta function, and the Dirichlet beta function. Full article

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

15 pages, 342 KiB

Open AccessArticle

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 1124

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

14 pages, 303 KiB

Open AccessArticle

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 1160

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

26 pages, 410 KiB

Open AccessArticle

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 1049

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

q_{1}

-type of the Hadamard–Kong product series is equal to zero if p Dirichlet series are of

q_{j}

-regular growth, and

q_{1} < q_{2} < \dots < q_{p}

,

q_{j} \in N_{+}

,

j = 1, 2, \dots, 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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

14 pages, 311 KiB

Open AccessArticle

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 1082

Abstract

We construct the Stancu-type generalization of q-Bernstein operators involving the idea of Bézier bases depending on the shape parameter

- 1 \leq ζ \leq 1

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

- 1 \leq ζ \leq 1

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

21 pages, 330 KiB

Open AccessArticle

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 1184

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

17 pages, 410 KiB

Open AccessArticle

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 1028

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

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10 pages, 264 KiB

Open AccessArticle

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 1196

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

20 pages, 868 KiB

Open AccessArticle

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 1435

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

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15 pages, 9978 KiB

Open AccessEditor’s ChoiceArticle

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 - 09 Aug 2021

Cited by 34 | Viewed by 2675

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

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14 pages, 315 KiB

Open AccessArticle

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 1537

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

15 pages, 822 KiB

Open AccessArticle

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 1291

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

18 pages, 326 KiB

Open AccessArticle

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 1669

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

12 pages, 264 KiB

Open AccessArticle

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 - 01 Sep 2020

Cited by 44 | Viewed by 1959

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

Review

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23 pages, 402 KiB

Open AccessReview

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 1389

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

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

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