Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 July 2024 | Viewed by 26245

Special Issue Editor

Department of Basic Sciences, Faculty of Engineering, Hasan Kalyoncu University, TR-27010 Gaziantep, Turkey
Interests: q-special functions and q-special polynomials; q-series; analytic number theory; umbral theory; p-adic q-analysis; fractional calculus and its applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Special functions and orthogonal polynomials, in particular, have been around for centuries. In the twentieth century, the emphasis was on special functions satisfying linear differential equations, but this has been extended to difference equations, partial differential equations and non-linear differential equations. The theory of the symmetries of special functions, orthogonal polynomials and differential equations is well improved, their relations to integrability are known, and there are many corresponding results and applications. They provide us the means to compute the symmetries of a given equation in an algorithmic manner and, most importantly, to implement it in symbolic computations.

This Special Issue will reflect the diversity of the topics across the world. The Special Issue’s papers will cover the symmetries of difference equations, discrete dynamical systems, special functions, orthogonal polynomials, symmetries, and integrable difference equations. The potential topics include but are not limited to the following:

  • Orthogonal polynomials;
  • Difference equations;
  • Symmetries in special functions;
  • Symmetries in orthogonal polynomials;
  • Symmetries of difference equations;
  • The analytical properties and applications of special functions;
  • Inequalities for special functions;
  • The integration of the products of special functions;
  • The properties of ordinary and general families of special polynomials;
  • Operational techniques involving special polynomials;
  • Classes of mixed special polynomials and their properties;
  • Other miscellaneous applications of special functions and special polynomials.

Dr. Serkan Araci
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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Published Papers (30 papers)

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Research

24 pages, 824 KiB  
Article
Geometric Properties of Normalized Galué Type Struve Function
Symmetry 2024, 16(2), 211; https://doi.org/10.3390/sym16020211 - 09 Feb 2024
Viewed by 500
Abstract
The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ, convexity of order δ, k-starlikeness, k-uniform [...] Read more.
The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ, convexity of order δ, k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature. Full article
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14 pages, 957 KiB  
Article
Functional and Operatorial Equations Defined Implicitly and Moment Problems
Symmetry 2024, 16(2), 152; https://doi.org/10.3390/sym16020152 - 27 Jan 2024
Viewed by 588
Abstract
The properties of the unique nontrivial analytic solution, defined implicitly by a functional equation, are pointed out. This work provides local estimations and global inequalities for the involved solution. The corresponding operatorial equation is studied as well. The second part of the paper [...] Read more.
The properties of the unique nontrivial analytic solution, defined implicitly by a functional equation, are pointed out. This work provides local estimations and global inequalities for the involved solution. The corresponding operatorial equation is studied as well. The second part of the paper is devoted to the full classical moment problem, which is an inverse problem. Two constraints are imposed on the solution. One of them requires the solution to be dominated by a concrete convex operator defined on the positive cone of the domain space. A one-dimensional operator is valued, and a multidimensional scalar moment problem is solved. In both cases, the existence and the uniqueness of the solution are proved. The general idea of the paper is to provide detailed information on solutions which are not expressible in terms of elementary functions. Full article
12 pages, 258 KiB  
Article
New Accurate Approximation Formula for Gamma Function
Symmetry 2024, 16(2), 150; https://doi.org/10.3390/sym16020150 - 27 Jan 2024
Viewed by 474
Abstract
In this paper, a new approximation formula for the gamma function and some of its symmetric inequalities are established. We prove the superiority of our results over Yang and Tian’s approximation formula for the gamma function of order v9. Full article
14 pages, 3651 KiB  
Article
The Properties of Topological Manifolds of Simplicial Polynomials
Symmetry 2024, 16(1), 102; https://doi.org/10.3390/sym16010102 - 14 Jan 2024
Viewed by 699
Abstract
The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions [...] Read more.
The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition. Full article
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12 pages, 1376 KiB  
Article
A Modified Residual Power Series Method for the Approximate Solution of Two-Dimensional Fractional Helmholtz Equations
Symmetry 2023, 15(12), 2152; https://doi.org/10.3390/sym15122152 - 04 Dec 2023
Viewed by 650
Abstract
In this paper, we suggest a modification for the residual power series method that is used to solve fractional-order Helmholtz equations, which is called the Shehu-transform residual power series method (ST-RPSM). This scheme uses a combination of the Shehu transform ( [...] Read more.
In this paper, we suggest a modification for the residual power series method that is used to solve fractional-order Helmholtz equations, which is called the Shehu-transform residual power series method (ST-RPSM). This scheme uses a combination of the Shehu transform (ST) and the residual power series method (RPSM). The fractional derivatives are taken with respect to Caputo order. The novelty of this approach is that it does not restrict the fractional order and reduces the need for heavy computational work. The results were obtained using an iterative series that led to an exact solution. The 3D graphical plots for different values of fractional orders are shown to compare ST-RPSM results with exact solutions. Full article
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17 pages, 2785 KiB  
Article
Computational Analysis on the Influence of Normal Force in a Homogeneous Isotropic Microstretch Thermoelastic Diffusive Solid
Symmetry 2023, 15(12), 2095; https://doi.org/10.3390/sym15122095 - 21 Nov 2023
Viewed by 559
Abstract
In this study, the identification of thermoelastic mass diffusion was examined on a homogeneous isotropic microstretch thermoelastic diffusion (HIMTD) solid due to normal force on the surface of half space. In the framework of Cartesian symmetry, the components of displacement, stresses, temperature change, [...] Read more.
In this study, the identification of thermoelastic mass diffusion was examined on a homogeneous isotropic microstretch thermoelastic diffusion (HIMTD) solid due to normal force on the surface of half space. In the framework of Cartesian symmetry, the components of displacement, stresses, temperature change, and microstretch as well as couple stress were investigated with and without microstretch and diffusion. The expression of the field functions was obtained using the Laplace and Fourier transforms. So as to estimate the nature of the components of displacement, stresses, temperature change, and microstretch as well as couple stress in the physical domain, an efficient approximate numerical inverse Laplace and Fourier transform technique and Romberg’s integration technique was adopted. It was meticulously considered and graphically illustrated how mass diffusion and microstretch affect thermoelastic deformation. Our objective was to address the inquiry regarding the impact of thermoelastic mass diffusion and microstretch on the field functions in the presence of a mass concentration source within the medium. Specifically, we aimed to investigate how these phenomena amplify the aforementioned effect. Full article
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18 pages, 333 KiB  
Article
Kneser-Type Oscillation Criteria for Half-Linear Delay Differential Equations of Third Order
Symmetry 2023, 15(11), 1994; https://doi.org/10.3390/sym15111994 - 29 Oct 2023
Viewed by 549
Abstract
This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, [...] Read more.
This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches the existing knowledge landscape within this field. One of the foundations on which we rely in proving the results is the symmetry between the positive and negative solutions, so that we can, using this feature, obtain criteria that guarantee the oscillation of all solutions. The paper enhances comprehension through the provision of illustrative examples that effectively showcase the outcomes and implications of the established findings. Full article
11 pages, 789 KiB  
Article
On Geometric Interpretations of Euler’s Substitutions
Symmetry 2023, 15(10), 1932; https://doi.org/10.3390/sym15101932 - 18 Oct 2023
Viewed by 816
Abstract
We consider a classical case of integrals containing an irrational integrand in the form of a square root of a quadratic polynomial. It is known that such “irrational integrals” can be expressed in terms of elementary functions by one of three of Euler’s [...] Read more.
We consider a classical case of integrals containing an irrational integrand in the form of a square root of a quadratic polynomial. It is known that such “irrational integrals” can be expressed in terms of elementary functions by one of three of Euler’s substitutions. It is less well known that the Euler substitutions have a geometric interpretation. In the framework of this interpretation, one can see that the number 3 is not the most suitable. We show that it is natural to introduce a fourth Euler substitution. In his original treatise, Leonhard Euler used two substitutions which are sufficient to cover all cases. Full article
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13 pages, 448 KiB  
Article
Solutions of Magnetohydrodynamics Equation through Symmetries
Symmetry 2023, 15(10), 1908; https://doi.org/10.3390/sym15101908 - 12 Oct 2023
Viewed by 819
Abstract
The magnetohydrodynamics (1 + 1) dimension equation, with a force and force-free term, is analysed with respect to its point symmetries. Interestingly, it reduces to an Abel’s Equation of the second kind and, under certain conditions, to equations specified in Gambier’s family. The [...] Read more.
The magnetohydrodynamics (1 + 1) dimension equation, with a force and force-free term, is analysed with respect to its point symmetries. Interestingly, it reduces to an Abel’s Equation of the second kind and, under certain conditions, to equations specified in Gambier’s family. The symmetry analysis for the force-free term leads to Euler’s Equation and to a system of reduced second-order odes for which singularity analysis is performed to determine their integrability. Full article
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19 pages, 364 KiB  
Article
Remarks on Approximate Solutions to Difference Equations in Various Spaces
Symmetry 2023, 15(10), 1829; https://doi.org/10.3390/sym15101829 - 26 Sep 2023
Viewed by 436
Abstract
Quite often (e.g., using numerical methods), we are only able to find approximate solutions of some equations, and it is necessary to know the size of the difference between such approximate solutions and the mappings that satisfy the equation exactly. This issue is [...] Read more.
Quite often (e.g., using numerical methods), we are only able to find approximate solutions of some equations, and it is necessary to know the size of the difference between such approximate solutions and the mappings that satisfy the equation exactly. This issue is the main subject of the theory of Ulam stability, and it is related to other areas of research such as, e.g., shadowing, optimization, and approximation theory. In this expository paper, we present several selected outcomes on Ulam stability of difference equations, show possible extensions of them and indicate further directions for research. We also present and discuss some simple methods that allow improvement of several already known results concerning Ulam stability of some difference equations in normed or metric spaces and extend them to b-metric and 2-normed spaces. Our results show that the noticeable symmetry exists between the outcomes of this type in normed and metric spaces and those obtained by us for other spaces. In particular, we extend the result of Pólya and Szegö concerning the stability of equation xn+m=xn+xm for m,nT, where T means either the set of integers Z or the set of positive integers N. We also consider the stability of equation xn+p+a1xn+p1++apxn+bn=0 (with a fixed positive integer p) and of two more general difference equations. Full article
14 pages, 279 KiB  
Article
Plane Partitions as Sums over Partitions
Symmetry 2023, 15(10), 1820; https://doi.org/10.3390/sym15101820 - 25 Sep 2023
Cited by 3 | Viewed by 838
Abstract
In this paper, we consider complete homogeneous symmetric functions and provide a new formula for the number of plane partitions of n. This formula expresses the number of plane partitions of n in terms of binomial coefficients as a sum over all [...] Read more.
In this paper, we consider complete homogeneous symmetric functions and provide a new formula for the number of plane partitions of n. This formula expresses the number of plane partitions of n in terms of binomial coefficients as a sum over all the partitions of n, considering the multiplicity of the parts greater than one. We obtain similar results for the number of strict plane partition of n and the number of symmetric plane partitions of n. Full article
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26 pages, 388 KiB  
Article
Generalized Polynomials and Their Unification and Extension to Discrete Calculus
Symmetry 2023, 15(9), 1677; https://doi.org/10.3390/sym15091677 - 31 Aug 2023
Viewed by 864
Abstract
In this paper, we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on (q;h)-time scales. Our proposed approach encompasses both difference and quantum problems, making it highly adoptable. [...] Read more.
In this paper, we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on (q;h)-time scales. Our proposed approach encompasses both difference and quantum problems, making it highly adoptable. Our framework employs forward and backward jump operators to create a unique approach. We use a weighted jump operator α that combines both jump operators in a convex manner. This allows us to generate a time scale α, which provides a new approach to discrete calculus. This beneficial approach enables us to define a general symmetric derivative on time scale α, which produces various types of discrete derivatives and forms a basis for new discrete calculus. Moreover, we create some polynomials on α-time scales using the α-operator. These polynomials have similar properties to regular polynomials and expand upon the existing research on discrete polynomials. Additionally, we establish the α-version of the Taylor formula. Finally, we discuss related binomial coefficients and their properties in discrete cases. We demonstrate how the symmetrical nature of the derivative definition allows for the incorporation of various concepts and the introduction of fresh ideas to discrete calculus. Full article
31 pages, 470 KiB  
Article
Applications of Euler Sums and Series Involving the Zeta Functions
Symmetry 2023, 15(9), 1637; https://doi.org/10.3390/sym15091637 - 24 Aug 2023
Viewed by 763
Abstract
A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we present several closed forms of series involving zeta functions by using [...] Read more.
A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis’s infinite product formula for π, Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums. Full article
21 pages, 857 KiB  
Article
Development of a Higher-Order 𝒜-Stable Block Approach with Symmetric Hybrid Points and an Adaptive Step-Size Strategy for Integrating Differential Systems Efficiently
Symmetry 2023, 15(9), 1635; https://doi.org/10.3390/sym15091635 - 24 Aug 2023
Viewed by 462
Abstract
This article introduces a computational hybrid one-step technique designed for solving initial value differential systems of a first order, which utilizes second derivative function evaluations. The method incorporates three intra-step symmetric points that are calculated to provide an optimum version of the suggested [...] Read more.
This article introduces a computational hybrid one-step technique designed for solving initial value differential systems of a first order, which utilizes second derivative function evaluations. The method incorporates three intra-step symmetric points that are calculated to provide an optimum version of the suggested scheme. By combining the hybrid and block methodologies, an efficient numerical method is achieved. The hybrid nature of the algorithm determines that the first Dahlquist barrier is overcome, ensuring its effectiveness. The proposed technique exhibits an eighth order of convergence and demonstrates A-stability characteristics, making it particularly well suited for handling stiff problems. Additionally, an adjustable step size variant of the algorithm is developed using an embedded-type technique. Through numerical experiments, it is shown that the suggested approach outperforms some other well-known methods with similar properties when applied to initial-value ordinary differential problems. Full article
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10 pages, 803 KiB  
Article
Vieta–Lucas Polynomials for the Brusselator System with the Rabotnov Fractional-Exponential Kernel Fractional Derivative
Symmetry 2023, 15(9), 1619; https://doi.org/10.3390/sym15091619 - 22 Aug 2023
Cited by 2 | Viewed by 698
Abstract
In this study, we provide an efficient simulation to investigate the behavior of the solution to the Brusselator system (a biodynamic system) with the Rabotnov fractional-exponential (RFE) kernel fractional derivative. A system of fractional differential equations can be used to represent this model. [...] Read more.
In this study, we provide an efficient simulation to investigate the behavior of the solution to the Brusselator system (a biodynamic system) with the Rabotnov fractional-exponential (RFE) kernel fractional derivative. A system of fractional differential equations can be used to represent this model. The fractional-order derivative of a polynomial function tp is approximated in terms of the RFE kernel. In this work, we employ shifted Vieta–Lucas polynomials in the spectral collocation technique. This process transforms the mathematical model into a set of algebraic equations. By assessing the residual error function, we can confirm that the provided approach is accurate and efficient. The outcomes demonstrate the effectiveness and simplicity of the technique for accurately simulating such models. Full article
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10 pages, 264 KiB  
Article
Symmetries of the Energy–Momentum Tensor for Static Plane Symmetric Spacetimes
Symmetry 2023, 15(8), 1614; https://doi.org/10.3390/sym15081614 - 21 Aug 2023
Viewed by 485
Abstract
This article explores matter collineations (MCs) of static plane-symmetric spacetimes, considering the stress–energy tensor in its contravariant and mixed forms. We solve the MC equations in two cases: when the energy–momentum tensor is nondegenerate and degenerate. For the case of a degenerate energy–momentum [...] Read more.
This article explores matter collineations (MCs) of static plane-symmetric spacetimes, considering the stress–energy tensor in its contravariant and mixed forms. We solve the MC equations in two cases: when the energy–momentum tensor is nondegenerate and degenerate. For the case of a degenerate energy–momentum tensor, we employ a direct integration technique to solve the MC equations, which leads to an infinite-dimensional Lie algebra. On the other hand, when considering the nondegenerate energy–momentum tensor, the contravariant form results in a finite-dimensional Lie algebra with dimensions of either 4 or 10. However, in the case of the mixed form of the energy–momentum tensor, the dimension of the Lie algebra is infinite. Moreover, the obtained MCs are compared with those already found for covariant stress–energy. Full article
19 pages, 283 KiB  
Article
Evaluation of the Poly-Jindalrae and Poly-Gaenari Polynomials in Terms of Degenerate Functions
Symmetry 2023, 15(8), 1587; https://doi.org/10.3390/sym15081587 - 15 Aug 2023
Cited by 1 | Viewed by 634
Abstract
The fundamental aim of this paper is to introduce the concept of poly-Jindalrae and poly-Gaenari numbers and polynomials within the context of degenerate functions. Furthermore, we give explicit expressions for these polynomial sequences and establish combinatorial identities that incorporate these polynomials. This includes [...] Read more.
The fundamental aim of this paper is to introduce the concept of poly-Jindalrae and poly-Gaenari numbers and polynomials within the context of degenerate functions. Furthermore, we give explicit expressions for these polynomial sequences and establish combinatorial identities that incorporate these polynomials. This includes the derivation of Dobinski-like formulas, recurrence relations, and other related aspects. Additionally, we present novel explicit expressions and identities of unipoly polynomials that are closely linked to some special numbers and polynomials. Full article
16 pages, 364 KiB  
Article
Symmetry Analysis for the 2D Aw-Rascle Traffic-Flow Model of Multi-Lane Motorways in the Euler and Lagrange Variables
Symmetry 2023, 15(8), 1525; https://doi.org/10.3390/sym15081525 - 02 Aug 2023
Cited by 1 | Viewed by 860
Abstract
A detailed symmetry analysis is performed for a microscopic model used to describe traffic flow in two-lane motorways. The traffic flow theory employed in this model is a two-dimensional extension of the Aw-Rascle theory. The flow parameters, including vehicle density, and vertical and [...] Read more.
A detailed symmetry analysis is performed for a microscopic model used to describe traffic flow in two-lane motorways. The traffic flow theory employed in this model is a two-dimensional extension of the Aw-Rascle theory. The flow parameters, including vehicle density, and vertical and horizontal velocities, are described by a system of first-order partial differential equations belonging to the family of hydrodynamic systems. This fluid-dynamics model is expressed in terms of the Euler and Lagrange variables. The admitted Lie point symmetries and the one-dimensional optimal system are determined for both sets of variables. It is found that the admitted symmetries for the two sets of variables form different Lie algebras, leading to distinct one-dimensional optimal systems. Finally, the Lie symmetries are utilized to derive new similarity closed-form solutions. Full article
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11 pages, 293 KiB  
Article
Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions
by
Symmetry 2023, 15(7), 1384; https://doi.org/10.3390/sym15071384 - 08 Jul 2023
Viewed by 688
Abstract
In this paper, we perform a further investigation on the Apostol–Bernoulli and Apostol–Euler functions introduced by Luo. By using the Fourier expansions of the Apostol–Bernoulli and Apostol–Euler polynomials, we establish some symmetric identities for the Apostol–Bernoulli and Apostol–Euler functions. As applications, some known [...] Read more.
In this paper, we perform a further investigation on the Apostol–Bernoulli and Apostol–Euler functions introduced by Luo. By using the Fourier expansions of the Apostol–Bernoulli and Apostol–Euler polynomials, we establish some symmetric identities for the Apostol–Bernoulli and Apostol–Euler functions. As applications, some known results, for example, Raabe’s multiplication formula and Hermite’s identity, are deduced as special cases. Full article
9 pages, 268 KiB  
Article
The Influence of the Perturbation of the Initial Data on the Analytic Approximate Solution of the Van der Pol Equation in the Complex Domain
Symmetry 2023, 15(6), 1200; https://doi.org/10.3390/sym15061200 - 03 Jun 2023
Cited by 3 | Viewed by 749
Abstract
In this paper, we substantiate the analytical approximate method for Cauchy problem of the Van der Pol equation in the complex domain. These approximate solutions allow analytical continuation for both real and complex cases. We follow the influence of variation in the initial [...] Read more.
In this paper, we substantiate the analytical approximate method for Cauchy problem of the Van der Pol equation in the complex domain. These approximate solutions allow analytical continuation for both real and complex cases. We follow the influence of variation in the initial data of the problem in order to control the computational process and improve the accuracy of the final results. Several simple applications of the method are given. A numerical study confirms the consistency of the developed method. Full article
13 pages, 290 KiB  
Article
On the Asymptotic Behavior of Class of Third-Order Neutral Differential Equations with Symmetrical and Advanced Argument
Symmetry 2023, 15(6), 1165; https://doi.org/10.3390/sym15061165 - 29 May 2023
Cited by 1 | Viewed by 662
Abstract
In this paper, we aimed to study some asymptotic properties of a class of third-order neutral differential equations with advanced argument in canonical form. We provide new and simplified oscillation criteria that improve and complement a number of existing results. We also show [...] Read more.
In this paper, we aimed to study some asymptotic properties of a class of third-order neutral differential equations with advanced argument in canonical form. We provide new and simplified oscillation criteria that improve and complement a number of existing results. We also show some examples to illustrate the importance of our results. Full article
9 pages, 279 KiB  
Article
Coefficient Bounds for Symmetric Subclasses of q-Convolution-Related Analytical Functions
Symmetry 2023, 15(6), 1133; https://doi.org/10.3390/sym15061133 - 23 May 2023
Viewed by 524
Abstract
By using q-convolution, we determine the coefficient bounds for certain symmetric subclasses of analytic functions of complex order, which are introduced here by means of a certain non-homogeneous Cauchy–Euler-type differential equation of order m. Full article
10 pages, 283 KiB  
Article
On Generalized Bivariate (p,q)-Bernoulli–Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli–Lucas Polynomials
Symmetry 2023, 15(4), 943; https://doi.org/10.3390/sym15040943 - 20 Apr 2023
Cited by 1 | Viewed by 791
Abstract
Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we define the generalized (p,q)-Bernoulli–Fibonacci [...] Read more.
Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we define the generalized (p,q)-Bernoulli–Fibonacci and generalized (p,q)-Bernoulli–Lucas polynomials and numbers by using the (p,q)-Bernoulli numbers, unified (p,q)-Bernoulli polynomials, h(x)-Fibonacci polynomials, and h(x)-Lucas polynomials. We also introduce the generalized bivariate (p,q)-Bernoulli–Fibonacci and generalized bivariate (p,q)-Bernoulli–Lucas polynomials and numbers. Then, we derive some properties of these newly established polynomials and numbers by using their generating functions with their functional equations. Finally, we provide some families of bilinear and bilateral generating functions for the generalized bivariate (p,q)-Bernoulli–Fibonacci polynomials. Full article
9 pages, 259 KiB  
Article
On Some Bounds for the Gamma Function
Symmetry 2023, 15(4), 937; https://doi.org/10.3390/sym15040937 - 19 Apr 2023
Cited by 1 | Viewed by 778
Abstract
Both theoretical and applied mathematics depend heavily on inequalities, which are rich in symmetries. In numerous studies, estimations of various functions based on the characteristics of their symmetry have been provided through inequalities. In this paper, we study the monotonicity of certain functions [...] Read more.
Both theoretical and applied mathematics depend heavily on inequalities, which are rich in symmetries. In numerous studies, estimations of various functions based on the characteristics of their symmetry have been provided through inequalities. In this paper, we study the monotonicity of certain functions that involve Gamma functions. We were able to obtain some of the bounds of Γ(v) that are more accurate than some recently published inequalities. Full article
15 pages, 340 KiB  
Article
Bounds for Extreme Zeros of Classical Orthogonal Polynomials Related to Birth and Death Processes
Symmetry 2023, 15(4), 890; https://doi.org/10.3390/sym15040890 - 10 Apr 2023
Viewed by 807
Abstract
In this paper, we consider birth and death processes with different sequences of transition rates and find the bound for the extreme zeros of orthogonal polynomials related to the three term recurrence relations and birth and death processes. Furthermore, we find the related [...] Read more.
In this paper, we consider birth and death processes with different sequences of transition rates and find the bound for the extreme zeros of orthogonal polynomials related to the three term recurrence relations and birth and death processes. Furthermore, we find the related chain sequences. Using these chain sequences, we find the transition probabilities for the corresponding process. As a consequence, transition probabilities related to G-fractions and modular forms are derived. Results obtained in this work are new and several graphical representations and numerical computations are provided to validate the results. Full article
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24 pages, 1495 KiB  
Article
Asymptotic Approximations of Higher-Order Apostol–Frobenius–Genocchi Polynomials with Enlarged Region of Validity
Symmetry 2023, 15(4), 876; https://doi.org/10.3390/sym15040876 - 06 Apr 2023
Viewed by 1229
Abstract
In this paper, the uniform approximations of the Apostol–Frobenius–Genocchi polynomials of order α in terms of the hyperbolic functions are derived through the saddle-point method. Moreover, motivated by the works of Corcino et al., an approximation with enlarged region of validity for these [...] Read more.
In this paper, the uniform approximations of the Apostol–Frobenius–Genocchi polynomials of order α in terms of the hyperbolic functions are derived through the saddle-point method. Moreover, motivated by the works of Corcino et al., an approximation with enlarged region of validity for these polynomials is also obtained. It is found out that the methods are also applicable for the case of the higher order generalized Apostol-type Frobenius–Genocchi polynomials and Apostol–Frobenius-type poly-Genocchi polynomials with parameters a, b, and c. These methods demonstrate the techniques of computing the symmetries of the defining equation of these polynomials. Graphs are illustrated to show the accuracy of the exact values and corresponding approximations of these polynomials with respect to some specific values of its parameters. Full article
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14 pages, 314 KiB  
Article
On (p,q)–Fibonacci and (p,q)–Lucas Polynomials Associated with Changhee Numbers and Their Properties
Symmetry 2023, 15(4), 851; https://doi.org/10.3390/sym15040851 - 02 Apr 2023
Cited by 2 | Viewed by 888
Abstract
Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties have been studied in the literature with the help of generating functions and their functional equations. In this paper, using the (p,q)–Fibonacci polynomials,  [...] Read more.
Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties have been studied in the literature with the help of generating functions and their functional equations. In this paper, using the (p,q)–Fibonacci polynomials, (p,q)–Lucas polynomials, and Changhee numbers, we define the (p,q)–Fibonacci–Changhee polynomials and (p,q)–Lucas–Changhee polynomials, respectively. We obtain some important identities and relations of these newly established polynomials by using their generating functions and functional equations. Then, we generalize the (p,q)–Fibonacci–Changhee polynomials and the (p,q)–Lucas–Changhee polynomials called generalized (p,q)–Fibonacci–Lucas–Changhee polynomials. We derive a determinantal representation for the generalized (p,q)–Fibonacci–Lucas–Changhee polynomials in terms of the special Hessenberg determinant. Finally, we give a new recurrent relation of the (p,q)–Fibonacci–Lucas–Changhee polynomials. Full article
20 pages, 500 KiB  
Article
Approximate and Exact Solutions in the Sense of Conformable Derivatives of Quantum Mechanics Models Using a Novel Algorithm
Symmetry 2023, 15(3), 744; https://doi.org/10.3390/sym15030744 - 17 Mar 2023
Cited by 15 | Viewed by 1444
Abstract
The entirety of the information regarding a subatomic particle is encoded in a wave function. Solving quantum mechanical models (QMMs) means finding the quantum mechanical wave function. Therefore, great attention has been paid to finding solutions for QMMs. In this study, a novel [...] Read more.
The entirety of the information regarding a subatomic particle is encoded in a wave function. Solving quantum mechanical models (QMMs) means finding the quantum mechanical wave function. Therefore, great attention has been paid to finding solutions for QMMs. In this study, a novel algorithm that combines the conformable Shehu transform and the Adomian decomposition method is presented that establishes approximate and exact solutions to QMMs in the sense of conformable derivatives with zero and nonzero trapping potentials. This solution algorithm is known as the conformable Shehu transform decomposition method (CSTDM). To evaluate the efficiency of this algorithm, the numerical results in terms of absolute and relative errors were compared with the reduced differential transform and the two-dimensional differential transform methods. The comparison showed excellent agreement with these methods, which means that the CSTDM is a suitable alternative tool to the methods based on the Caputo derivative for the solutions of time-fractional QMMs. The advantage of employing this approach is that, due to the use of the conformable Shehu transform, the pattern between the coefficients of the series solutions makes it simple to obtain the exact solution of both linear and nonlinear problems. Consequently, our approach is quick, accurate, and easy to implement. The convergence, uniqueness, and error analysis of the solution were examined using Banach’s fixed point theory. Full article
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11 pages, 278 KiB  
Article
An Application of Poisson Distribution Series on Harmonic Classes of Analytic Functions
Symmetry 2023, 15(3), 590; https://doi.org/10.3390/sym15030590 - 24 Feb 2023
Cited by 4 | Viewed by 963
Abstract
Many authors have obtained some inclusion properties of certain subclasses of univalent and functions associated with distribution series, such as Pascal distribution, Binomial distribution, Poisson distribution, Mittag–Leffler-type Poisson distribution, and Geometric distribution. In the present paper, we obtain some inclusion relations of the [...] Read more.
Many authors have obtained some inclusion properties of certain subclasses of univalent and functions associated with distribution series, such as Pascal distribution, Binomial distribution, Poisson distribution, Mittag–Leffler-type Poisson distribution, and Geometric distribution. In the present paper, we obtain some inclusion relations of the harmonic class H(α,δ) with the classes SH* of starlike harmonic functions and KH of convex harmonic functions, also for the harmonic classes TNHβ and TRHβ associated with the operator Υ defined by applying certain convolution operator regarding Poisson distribution series. Several consequences and corollaries of the main results are also obtained. Full article
21 pages, 380 KiB  
Article
Potentials from the Polynomial Solutions of the Confluent Heun Equation
Symmetry 2023, 15(2), 461; https://doi.org/10.3390/sym15020461 - 09 Feb 2023
Cited by 3 | Viewed by 1507
Abstract
Polynomial solutions of the confluent Heun differential equation (CHE) are derived by identifying conditions under which the infinite power series expansions around the z=0 singular point can be terminated. Assuming a specific structure of the expansion coefficients, these conditions lead to [...] Read more.
Polynomial solutions of the confluent Heun differential equation (CHE) are derived by identifying conditions under which the infinite power series expansions around the z=0 singular point can be terminated. Assuming a specific structure of the expansion coefficients, these conditions lead to four non-trivial polynomials that can be expressed as special cases of the confluent Heun function Hc(p,β,γ,δ,σ;z). One of these recovers the generalized Laguerre polynomials LN(α), and another one the rationally extended X1 type Laguerre polynomials L^N(α). The two remaining solutions represent previously unknown polynomials that do not form an orthogonal set and exhibit features characteristic of semi-classical orthogonal polynomials. A standard method of generating exactly solvable potentials in the one-dimensional Schrödinger equation is applied to the CHE, and all known potentials with solutions expressed in terms of the generalized Laguerre polynomials within, or outside the Natanzon confluent potential class, are recovered. It is also found that the potentials generated from the two new polynomial systems necessarily depend on the N quantum number. General considerations on the application of the Heun type differential differential equations within the present framework are also discussed. Full article
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