Symmetry

Research

15 pages, 287 KiB

Open AccessArticle

A Uniqueness Theorem for Stability Problems of Functional Equations

by Soon-Mo Jung, Yang-Hi Lee and Jaiok Roh

Symmetry 2024, 16(10), 1298; https://doi.org/10.3390/sym16101298 - 2 Oct 2024

Viewed by 363

In this paper, we present a uniqueness theorem obtained by using direct calculation. This theorem is applicable to stability problems of functional equations whose solutions are monomial or generalized polynomial mappings of degree n. The advantage of this uniqueness theorem is that [...] Read more.

In this paper, we present a uniqueness theorem obtained by using direct calculation. This theorem is applicable to stability problems of functional equations whose solutions are monomial or generalized polynomial mappings of degree n. The advantage of this uniqueness theorem is that it simplifies the proof by eliminating the need to repeatedly and cumbersomely prove uniqueness in stability studies. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

24 pages, 992 KiB

Open AccessArticle

Fractional-Order Correlation between Special Functions Inspired by Bone Fractal Operators

by Zhimo Jian, Chaoqian Luo, Tianyi Zhou, Gang Peng and Yajun Yin

Symmetry 2024, 16(10), 1279; https://doi.org/10.3390/sym16101279 - 29 Sep 2024

Viewed by 830

Abstract

In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be characterized by fractal operators, and each type of operator possesses rich invariant properties. [...] Read more.

In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be characterized by fractal operators, and each type of operator possesses rich invariant properties. Based on the invariant properties of fractal operators, we discovered that the symmetry evolution laws of functional fractal trees in the physical fractal space can reveal the intrinsic correlations between special functions. This article explores the fractional-order correlation between special functions inspired by bone fractal operators. Specifically, the following contents are included: (1) showing the intrinsic expression in the convolutional kernel function of bone fractal operators and its correlation with special functions; (2) proving the following proposition: the convolutional kernel function of bone fractal operators is still related to the special functions under different input signals (external load, external stimulus); (3) using the bone fractal operators as the background and error function as the core, deriving the fractional-order correlation between different special functions. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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15 pages, 300 KiB

Open AccessArticle

Some Properties on Normalized Tails of Maclaurin Power Series Expansion of Exponential Function

by Zhi-Hua Bao, Ravi Prakash Agarwal, Feng Qi and Wei-Shih Du

Symmetry 2024, 16(8), 989; https://doi.org/10.3390/sym16080989 - 5 Aug 2024

Cited by 2 | Viewed by 1156

Abstract

In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of [...] Read more.

In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of an elementary function involving the exponential function, (4) with the help of an integral representation for the tail of the power series expansion of the exponential function, and (5) on account of Čebyšev’s integral inequality, the authors (i) expand the logarithm of the normalized tail of the power series expansion of the exponential function into a power series whose coefficients are expressed in terms of specific Hessenberg determinants whose elements are quotients of combinatorial numbers, (ii) prove the logarithmic convexity of the normalized tail of the power series expansion of the exponential function, (iii) derive a new determinantal expression of the Bernoulli numbers, deduce a determinantal expression for Howard’s numbers, (iv) confirm the increasing monotonicity of a function related to the logarithm of the normalized tail of the power series expansion of the exponential function, (v) present an inequality among three power series whose coefficients are reciprocals of combinatorial numbers, and (vi) generalize the logarithmic convexity of an extensively applied function involving the exponential function. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

12 pages, 1495 KiB

Open AccessArticle

Geometric Features of the Hurwitz–Lerch Zeta Type Function Based on Differential Subordination Method

by Faten F. Abdulnabi, Hiba F. Al-Janaby, Firas Ghanim and Alina Alb Lupaș

Symmetry 2024, 16(7), 784; https://doi.org/10.3390/sym16070784 - 21 Jun 2024

Cited by 1 | Viewed by 1091

Abstract

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine [...] Read more.

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine the geometric series, logarithm, and Hurwitz–Lerch zeta functions to formulate a new special function, namely, the logarithm-Hurwitz–Lerch zeta function (LHL-Z function). This investigation then contributes to the study of the LHL-Z function in terms of the geometric theory of holomorphic functions, based on the differential subordination methodology, to discuss and determine the univalence and convexity conditions of the LHL-Z function. Moreover, there are other subordination and superordination connections that may be visually represented using geometric methods. Functions often exhibit symmetry when subjected to conformal mappings. The investigation of the symmetries of these mappings may provide a clearer understanding of how subordination and superordination with the Hurwitz–Lerch zeta function behave under different transformations. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

20 pages, 280 KiB

Open AccessArticle

On the Inverse of the Linearization Coefficients of Bessel Polynomials

by Mohamed Jalel Atia

Symmetry 2024, 16(6), 737; https://doi.org/10.3390/sym16060737 - 13 Jun 2024

Viewed by 539

Abstract

In this contribution, we first present a new recursion relation fulfilled by the linearization coefficients of Bessel polynomials (LCBPs), which is different than the one presented by Berg and Vignat in 2008. We will explain why this new recursion formula is as important [...] Read more.

In this contribution, we first present a new recursion relation fulfilled by the linearization coefficients of Bessel polynomials (LCBPs), which is different than the one presented by Berg and Vignat in 2008. We will explain why this new recursion formula is as important as Berg and Vignat’s. We introduce the matrix linearization coefficients of Bessel polynomials (MLCBPs), and we present some new results and some conjectures on these matrices. Second, we present the inverse of the connection coefficients with an application involving the modified Bessel function of the second kind. Finally, we introduce the inverse of the matrix of the linearization coefficients of the Bessel polynomials (IMLCBPs), and we present some open problems related to these IMLCBPs. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

11 pages, 264 KiB

Open AccessArticle

Several Symmetric Identities of the Generalized Degenerate Fubini Polynomials by the Fermionic p-Adic Integral on

Z_{p}

by Maryam Salem Alatawi, Waseem Ahmad Khan and Ugur Duran

Symmetry 2024, 16(6), 686; https://doi.org/10.3390/sym16060686 - 3 Jun 2024

Viewed by 417

Abstract

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of [...] Read more.

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of many families of special polynomials and numbers, such as Bernoulli, Fubini, Bell, and Changhee polynomials and numbers. One of the main applications of these integrals is to obtain symmetric identities for the special polynomials. In this study, we focus on a novel extension of the degenerate Fubini polynomials and on obtaining some symmetric identities for them. First, we introduce the two-variable degenerate w-torsion Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. By this representation, we derive some new symmetric identities for these polynomials, using some special p-adic integral techniques. Lastly, by using some series manipulation techniques, we obtain more identities of symmetry for the two variable degenerate w-torsion Fubini polynomials. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

16 pages, 355 KiB

Open AccessArticle

Sufficient Conditions for Linear Operators Related to Confluent Hypergeometric Function and Generalized Bessel Function of the First Kind to Belong to a Certain Class of Analytic Functions

by Saiful R. Mondal, Manas Kumar Giri and Raghavendar Kondooru

Symmetry 2024, 16(6), 662; https://doi.org/10.3390/sym16060662 - 27 May 2024

Cited by 1 | Viewed by 581

Abstract

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are [...] Read more.

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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16 pages, 5574 KiB

Open AccessArticle

Topological Deformations of Manifolds by Algebraic Compositions in Polynomial Rings

by Susmit Bagchi

Symmetry 2024, 16(5), 556; https://doi.org/10.3390/sym16050556 - 3 May 2024

Viewed by 1238

Abstract

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending [...] Read more.

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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12 pages, 268 KiB

Open AccessArticle

Some New Fractional Inequalities Defined Using cr-Log-h-Convex Functions and Applications

by Sikander Mehmood, Pshtiwan Othman Mohammed, Artion Kashuri, Nejmeddine Chorfi, Sarkhel Akbar Mahmood and Majeed A. Yousif

Symmetry 2024, 16(4), 407; https://doi.org/10.3390/sym16040407 - 1 Apr 2024

Cited by 4 | Viewed by 1429

Abstract

There is a strong correlation between the concept of convexity and symmetry. One of these is the class of interval-valued cr-log-h-convex functions, which is closely related to the theory of symmetry. In this paper, we obtain Hermite–Hadamard and its weighted version inequalities that [...] Read more.

There is a strong correlation between the concept of convexity and symmetry. One of these is the class of interval-valued cr-log-h-convex functions, which is closely related to the theory of symmetry. In this paper, we obtain Hermite–Hadamard and its weighted version inequalities that are related to interval-valued cr-log-h-convex functions, and some known results are recaptured. To support our main results, we offer three examples and two applications related to modified Bessel functions and special means as well. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

14 pages, 267 KiB

Open AccessArticle

Some Properties of a Falling Function and Related Inequalities on Green’s Functions

by Pshtiwan Othman Mohammed, Ravi P. Agarwal, Majeed A. Yousif, Eman Al-Sarairah, Sarkhel Akbar Mahmood and Nejmeddine Chorfi

Symmetry 2024, 16(3), 337; https://doi.org/10.3390/sym16030337 - 11 Mar 2024

Cited by 3 | Viewed by 1258

Abstract

Asymmetry plays a significant role in the transmission dynamics in novel discrete fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed discrete models that incorporate different symmetry developmental stages. This paper introduces a Taylor monomial falling function and [...] Read more.

Asymmetry plays a significant role in the transmission dynamics in novel discrete fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed discrete models that incorporate different symmetry developmental stages. This paper introduces a Taylor monomial falling function and presents some properties of this function in a delta fractional model with Green’s function kernel. In the deterministic case, Green’s function will be non-negative, and this shows that the function has an upper bound for its maximum point. More precisely, in this paper, based on the properties of the Taylor monomial falling function, we investigate Lyapunov-type inequalities for a delta fractional boundary value problem of Riemann–Liouville type. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

42 pages, 3002 KiB

Open AccessArticle

Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials

by Gregory Natanson

Symmetry 2024, 16(3), 282; https://doi.org/10.3390/sym16030282 - 29 Feb 2024

Viewed by 925

Abstract

This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by [...] Read more.

This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

24 pages, 824 KiB

Open AccessArticle

Geometric Properties of Normalized Galué Type Struve Function

by Samanway Sarkar, Sourav Das and Saiful R. Mondal

Symmetry 2024, 16(2), 211; https://doi.org/10.3390/sym16020211 - 9 Feb 2024

Viewed by 1078

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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14 pages, 957 KiB

Open AccessArticle

Functional and Operatorial Equations Defined Implicitly and Moment Problems

by Octav Olteanu

Symmetry 2024, 16(2), 152; https://doi.org/10.3390/sym16020152 - 27 Jan 2024

Cited by 1 | Viewed by 1002

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

12 pages, 258 KiB

Open AccessArticle

New Accurate Approximation Formula for Gamma Function

by Mansour Mahmoud and Hanan Almuashi

Symmetry 2024, 16(2), 150; https://doi.org/10.3390/sym16020150 - 27 Jan 2024

Viewed by 1317

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

v^{- 9}

. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

14 pages, 3651 KiB

Open AccessArticle

The Properties of Topological Manifolds of Simplicial Polynomials

by Susmit Bagchi

Symmetry 2024, 16(1), 102; https://doi.org/10.3390/sym16010102 - 14 Jan 2024

Cited by 1 | Viewed by 1272

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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12 pages, 1376 KiB

Open AccessArticle

A Modified Residual Power Series Method for the Approximate Solution of Two-Dimensional Fractional Helmholtz Equations

by Jinxing Liu, Muhammad Nadeem, Asad Islam, Sorin Mureşan and Loredana Florentina Iambor

Symmetry 2023, 15(12), 2152; https://doi.org/10.3390/sym15122152 - 4 Dec 2023

Cited by 1 | Viewed by 1227

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 (

S

T-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 (

S

T-RPSM). This scheme uses a combination of the Shehu transform (

S

T) 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

S

T-RPSM results with exact solutions. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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17 pages, 2785 KiB

Open AccessArticle

Computational Analysis on the Influence of Normal Force in a Homogeneous Isotropic Microstretch Thermoelastic Diffusive Solid

by Kulvinder Singh, Iqbal Kaur and Marin Marin

Symmetry 2023, 15(12), 2095; https://doi.org/10.3390/sym15122095 - 21 Nov 2023

Viewed by 857

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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18 pages, 333 KiB

Open AccessArticle

Kneser-Type Oscillation Criteria for Half-Linear Delay Differential Equations of Third Order

by Fahd Masood, Clemente Cesarano, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani and Hamdy El-Metwally

Symmetry 2023, 15(11), 1994; https://doi.org/10.3390/sym15111994 - 29 Oct 2023

Cited by 4 | Viewed by 1021

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

11 pages, 789 KiB

Open AccessArticle

On Geometric Interpretations of Euler’s Substitutions

by Jan L. Cieśliński and Maciej Jurgielewicz

Symmetry 2023, 15(10), 1932; https://doi.org/10.3390/sym15101932 - 18 Oct 2023

Viewed by 1533

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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13 pages, 448 KiB

Open AccessArticle

Solutions of Magnetohydrodynamics Equation through Symmetries

by Rangasamy Sinuvasan, Amlan K. Halder, Rajeswari Seshadri, Andronikos Paliathanasis and Peter G. L. Leach

Symmetry 2023, 15(10), 1908; https://doi.org/10.3390/sym15101908 - 12 Oct 2023

Viewed by 1411

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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19 pages, 364 KiB

Open AccessArticle

Remarks on Approximate Solutions to Difference Equations in Various Spaces

by Janusz Brzdęk

Symmetry 2023, 15(10), 1829; https://doi.org/10.3390/sym15101829 - 26 Sep 2023

Viewed by 895

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

x_{n + m} = x_{n} + x_{m}

for

m, n \in T

, where T means either the set of integers

Z

or the set of positive integers

N

. We also consider the stability of equation

x_{n + p} + a_{1} x_{n + p - 1} + \dots + a_{p} x_{n} + b_{n} = 0

(with a fixed positive integer p) and of two more general difference equations. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

14 pages, 279 KiB

Open AccessArticle

Plane Partitions as Sums over Partitions

by Mircea Merca and Iulia-Ionelia Radu

Symmetry 2023, 15(10), 1820; https://doi.org/10.3390/sym15101820 - 25 Sep 2023

Cited by 3 | Viewed by 1339

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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26 pages, 388 KiB

Open AccessArticle

Generalized Polynomials and Their Unification and Extension to Discrete Calculus

by Mieczysław Cichoń, Burcu Silindir, Ahmet Yantir and Seçil Gergün

Symmetry 2023, 15(9), 1677; https://doi.org/10.3390/sym15091677 - 31 Aug 2023

Cited by 1 | Viewed by 1409

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

31 pages, 470 KiB

Open AccessArticle

Applications of Euler Sums and Series Involving the Zeta Functions

by Junesang Choi and Anthony Sofo

Symmetry 2023, 15(9), 1637; https://doi.org/10.3390/sym15091637 - 24 Aug 2023

Viewed by 1406

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

21 pages, 857 KiB

Open AccessArticle

Development of a Higher-Order 𝒜-Stable Block Approach with Symmetric Hybrid Points and an Adaptive Step-Size Strategy for Integrating Differential Systems Efficiently

by Rajat Singla, Gurjinder Singh, Higinio Ramos and Vinay Kanwar

Symmetry 2023, 15(9), 1635; https://doi.org/10.3390/sym15091635 - 24 Aug 2023

Viewed by 836

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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10 pages, 803 KiB

Open AccessArticle

Vieta–Lucas Polynomials for the Brusselator System with the Rabotnov Fractional-Exponential Kernel Fractional Derivative

by Mohamed M. Khader, Jorge E. Macías-Díaz, Khaled M. Saad and Waleed M. Hamanah

Symmetry 2023, 15(9), 1619; https://doi.org/10.3390/sym15091619 - 22 Aug 2023

Cited by 5 | Viewed by 1158

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

t^{p}

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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10 pages, 264 KiB

Open AccessArticle

Symmetries of the Energy–Momentum Tensor for Static Plane Symmetric Spacetimes

by Fawad Khan, Wajid Ullah, Tahir Hussain and Wojciech Sumelka

Symmetry 2023, 15(8), 1614; https://doi.org/10.3390/sym15081614 - 21 Aug 2023

Viewed by 950

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

19 pages, 283 KiB

Open AccessArticle

Evaluation of the Poly-Jindalrae and Poly-Gaenari Polynomials in Terms of Degenerate Functions

by Noor Alam, Waseem Ahmad Khan, Serkan Araci, Hasan Nihal Zaidi and Anas Al Taleb

Symmetry 2023, 15(8), 1587; https://doi.org/10.3390/sym15081587 - 15 Aug 2023

Cited by 2 | Viewed by 978

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

16 pages, 364 KiB

Open AccessArticle

Symmetry Analysis for the 2D Aw-Rascle Traffic-Flow Model of Multi-Lane Motorways in the Euler and Lagrange Variables

by Andronikos Paliathanasis

Symmetry 2023, 15(8), 1525; https://doi.org/10.3390/sym15081525 - 2 Aug 2023

Cited by 3 | Viewed by 1448

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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11 pages, 293 KiB

Open AccessArticle

Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions

by Yuan He

Symmetry 2023, 15(7), 1384; https://doi.org/10.3390/sym15071384 - 8 Jul 2023

Viewed by 1113

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

9 pages, 268 KiB

Open AccessArticle

The Influence of the Perturbation of the Initial Data on the Analytic Approximate Solution of the Van der Pol Equation in the Complex Domain

by Victor Orlov and Alexander Chichurin

Symmetry 2023, 15(6), 1200; https://doi.org/10.3390/sym15061200 - 3 Jun 2023

Cited by 3 | Viewed by 1094

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

13 pages, 290 KiB

Open AccessArticle

On the Asymptotic Behavior of Class of Third-Order Neutral Differential Equations with Symmetrical and Advanced Argument

by Munirah Aldiaiji, Belgees Qaraad, Loredana Florentina Iambor and Elmetwally M. Elabbasy

Symmetry 2023, 15(6), 1165; https://doi.org/10.3390/sym15061165 - 29 May 2023

Cited by 2 | Viewed by 1082

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

9 pages, 279 KiB

Open AccessArticle

Coefficient Bounds for Symmetric Subclasses of q-Convolution-Related Analytical Functions

by Sheza M. El-Deeb and Luminita-Ioana Cotîrlă

Symmetry 2023, 15(6), 1133; https://doi.org/10.3390/sym15061133 - 23 May 2023

Viewed by 893

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

10 pages, 283 KiB

Open AccessArticle

On Generalized Bivariate (p,q)-Bernoulli–Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli–Lucas Polynomials

by Hao Guan, Waseem Ahmad Khan and Can Kızılateş

Symmetry 2023, 15(4), 943; https://doi.org/10.3390/sym15040943 - 20 Apr 2023

Cited by 5 | Viewed by 1300

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

9 pages, 259 KiB

Open AccessArticle

On Some Bounds for the Gamma Function

by Mansour Mahmoud, Saud M. Alsulami and Safiah Almarashi

Symmetry 2023, 15(4), 937; https://doi.org/10.3390/sym15040937 - 19 Apr 2023

Cited by 2 | Viewed by 1210

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

15 pages, 340 KiB

Open AccessArticle

Bounds for Extreme Zeros of Classical Orthogonal Polynomials Related to Birth and Death Processes

by Saiful R. Mondal and Sourav Das

Symmetry 2023, 15(4), 890; https://doi.org/10.3390/sym15040890 - 10 Apr 2023

Viewed by 1159

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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24 pages, 1495 KiB

Open AccessArticle

Asymptotic Approximations of Higher-Order Apostol–Frobenius–Genocchi Polynomials with Enlarged Region of Validity

by Cristina Corcino, Wilson D. Castañeda, Jr. and Roberto Corcino

Symmetry 2023, 15(4), 876; https://doi.org/10.3390/sym15040876 - 6 Apr 2023

Viewed by 1695

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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14 pages, 314 KiB

Open AccessArticle

On (p,q)–Fibonacci and (p,q)–Lucas Polynomials Associated with Changhee Numbers and Their Properties

by Chuanjun Zhang, Waseem Ahmad Khan and Can Kızılateş

Symmetry 2023, 15(4), 851; https://doi.org/10.3390/sym15040851 - 2 Apr 2023

Cited by 5 | Viewed by 1451

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

20 pages, 500 KiB

Open AccessArticle

Approximate and Exact Solutions in the Sense of Conformable Derivatives of Quantum Mechanics Models Using a Novel Algorithm

by Muhammad Imran Liaqat, Ali Akgül, Manuel De la Sen and Mustafa Bayram

Symmetry 2023, 15(3), 744; https://doi.org/10.3390/sym15030744 - 17 Mar 2023

Cited by 23 | Viewed by 2091

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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11 pages, 278 KiB

Open AccessArticle

An Application of Poisson Distribution Series on Harmonic Classes of Analytic Functions

by Basem Frasin and Alina Alb Lupaş

Symmetry 2023, 15(3), 590; https://doi.org/10.3390/sym15030590 - 24 Feb 2023

Cited by 5 | Viewed by 1465

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

S_{H}^{*}

of starlike harmonic functions and

K_{H}

of convex harmonic functions, also for the harmonic classes

{TN}_{H} (β)

and

{TR}_{H} (β)

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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

21 pages, 380 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Potentials from the Polynomial Solutions of the Confluent Heun Equation

by Géza Lévai

Symmetry 2023, 15(2), 461; https://doi.org/10.3390/sym15020461 - 9 Feb 2023

Cited by 5 | Viewed by 2354

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

H c (p, β, γ, δ, σ; z)

. One of these recovers the generalized Laguerre polynomials

L_{N}^{(α)}

, and another one the rationally extended

X_{1}

type Laguerre polynomials

{\hat{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

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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