Gegenbauer Polynomials Research Articles

Orthogonal Polynomials and Sharp Estimates for the Schrödinger Equation

In this article, we study sharp estimates for the Schrödinger operator via the framework of orthogonal polynomials. We use Hermite and Laguerre polynomial expansions to produce sharp Strichartz estimates for even exponents. In particular, for radial initial data in dimension |$2$|⁠, we establish an interesting connection of the Strichartz norm with a combinatorial problem about words with four letters. We use spherical harmonics and Gegenbauer polynomials to prove a sharpened weighted inequality for the Schrödinger equation that is maximized by radial functions.

Matrix-Valued Gegenbauer-Type Polynomials

We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter nu >0. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters nu and nu +1. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauer-type polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case nu =1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.

Plane wave formulas for spherical, complex and symplectic harmonics

This paper is concerned with spherical harmonics, and two refinements thereof: complex harmonics and symplectic harmonics.The reproducing kernels of the spherical and complex harmonics are explicitly given in terms of Gegenbauer or Jacobi polynomials. In the first part of the paper we determine the reproducing kernel for the space of symplectic harmonics, which is again expressible as a Jacobi polynomial of a suitable argument.In the second part we find plane wave formulas for the reproducing kernels of the three types of harmonics, expressing them as suitable integrals over Stiefel manifolds. This is achieved using Pizzetti formulas that express the integrals in terms of differential operators.

Computation of Hyperspherical Bessel Functions

AbstractIn this paper we present a fast and accurate numerical algorithm for the computation of hyperspherical Bessel functions of large order and real arguments. For the hyperspherical Bessel functions of closed type, no stable algorithm existed so far due to the lack of a backwards recurrence. We solved this problem by establishing a relation to Gegenbauer polynomials. All our algorithms are written in C and are publicly available at Github [https://github.com/lesgourg/class_public]. A Python wrapper is available upon request.

Quantum superintegrable Zernike system

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, whose value at the boundary can be nonzero. On this account, the quantum Zernike system, where that differential equation is seen as a Schrödinger equation with a potential, is special in that it has a potential and a boundary condition that are not standard in quantum mechanics. We project the disk on a half-sphere and there we find that, in addition to polar coordinates, this system separates into two additional coordinate systems (non-orthogonal on the pupil disk), which lead to Schrödinger-type equations with Pöschl-Teller potentials, whose eigen-solutions involve Legendre, Gegenbauer, and Jacobi polynomials. This provides new expressions for separated polynomial solutions of the original Zernike system that are real. The operators which provide the separation constants are found to participate in a superintegrable cubic Higgs algebra.

Some Ultraspheroidal Monogenic Clifford Gegenbauer Jacobi Polynomials and Associated Wavelets

In the present paper, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on 2-parameters weight functions. Such classes englobe the well known ones of Jacobi and Gegenbauer polynomials when relaxing one of the parameters. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula as well as Fourier-Plancherel rules have been proved.

Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

The determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states. For the D-dimensional hydrogenic and oscillator-like systems, the wavefunctions of the corresponding bound states are controlled by the Laguerre () and Gegenbauer () polynomials in both position and momentum spaces, where the parameter α linearly depends on D. In this work we study the asymptotic behavior as of the associated entropy-like integral functionals of these two families of hypergeometric polynomials.

Modified wavelets–based algorithm for nonlinear delay differential equations of fractional order

Most of the physical phenomena located around us are nonlinear in nature and their solutions are of great significance for scientists and engineers. In order to have a better representation of these physical phenomena, fractional calculus is developed. Some of these nonlinear physical models can be represented in the form of delay differential equations of fractional order. In this article, a new method named Gegenbauer Wavelets Steps Method is proposed using Gegenbauer polynomials and method of steps for solving nonlinear fractional delay differential equations. Method of steps is used to convert the fractional nonlinear fractional delay differential equation into a fractional nonlinear differential equation and then Gegenbauer wavelet method is applied at each iteration of fractional differential equation to find the solution. To check the accuracy and efficiency of the proposed method, the proposed method is implemented on different nonlinear fractional delay differential equations including singular-type problems also.

A Numerical Approach with Residual Error Estimation for Eolution of High-order Linear Differential-difference Equations by Using Gegenbauer Polynomials

The main aim of this study is to apply the Gegenbauer polynomials for the solution of high-order linear differential difference equations with functional arguments under initial-boundary conditions.The technique we have used is essentially based on the truncated Gegenbauer series and its matrix representations along with collocation points. Also, by using the Mean-Value Theorem and residual function, an efficient error estimation technique is proposed and some illustrative examples are presented to demonstrate the validity and applicability of the method.

On Coefficients Problems for Typically Real Functions Related to Gegenbauer Polynomials

We solve problems concerning the coefficients of functions in the class mathcal {T}(lambda ) of typically real functions associated with Gegenbauer polynomials. The main aim is to determine the estimates of two expressions: |a_4-a_2 a_3| and |a_2 a_4 -a_3{}^2|. The second one is known as the second Hankel determinant. In order to obtain these bounds, we consider the regions of variability of selected pairs of coefficients for functions in mathcal {T}(lambda ). Furthermore, we find the upper and the lower bounds of functionals of Fekete–Szegö type. Finally, we present some conclusions for the classes mathcal {T} and mathcal {T}(1/2).

Mitigating Gibbs Phenomena in Uncertainty Quantification With a Stochastic Spectral Method

The use of spectral projection-based methods for simulation of a stochastic system with discontinuous solution exhibits the Gibbs phenomenon, which is characterized by oscillations near discontinuities. This paper investigates a dynamic bi-orthogonality-based approach with appropriate postprocessing for mitigating the effects of the Gibbs phenomenon. The proposed approach uses spectral decomposition of the spatial and stochastic fields in appropriate orthogonal bases, while the dynamic orthogonality (DO) condition is used to derive the resultant closed-form evolution equations. The orthogonal decomposition of the spatial field is exploited to propose a Gegenbauer reprojection-based postprocessing approach, where the orthogonal bases in spatial dimension are reprojected on the Gegenbauer polynomials in the domain of analyticity. The resultant spectral expansion in Gegenbauer series is shown to mitigate the Gibbs phenomenon. Efficacy of the proposed method is demonstrated for simulation of a one-dimensional stochastic Burgers equation and stochastic quasi-one-dimensional flow through a convergent-divergent nozzle.

Flow generated by slow steady rotation of a permeable sphere in a micro-polar fluid

The analytical study of the flow generated by the slow steady rotation of a permeable sphere in an incompressible micro-polar fluid is considered. Both the flows internal and external to the sphere are coupled. The result will degenerate to independent equations for the case of viscous fluids for the inner and external flows. The flow field in the form of velocity w and micro rotation function φ are obtained in terms of modified Bessel functions and Gegenbauer polynomials. The flow pattern is shown in the form of graphs. It is interesting to note that the velocity and micro-rotation functions within the sphere are constant at distances from the axis of rotation since it represents a rigid body rotation. Effects of physical parameters on the Couple are also shown in the form of graphs.

On the Sobolev orthogonality of classical orthogonal polynomials for non standard parameters

The discrete part of the discrete-continuous orthogonality \[ \mathscr {B}(f,g)=\mathscr {B}_d( f,g)+\mathscr {B}_c(f^{(N)},g^{(N)}), \] is studied for families of classical orthogonal polynomials such that the associated three-term recurrence relation \[ xp_n=p_{n+1}+\beta _np_n+ \gamma _n p_{n-1}, \] presents one vanishing coefficient $\gamma _n$, as in the case of Laguerre polynomials $L_n^{(-N)}$, Jacobi polynomials $P_n^{(-N,\beta )}$ and Gegenbauer polynomials $C_n^{(-N+1/2)}$ with $N\in \mathbb {N}$. It is shown that the discrete bilinear functional $\mathscr {B}_d$ can be replaced by a linear functional, $\mathscr {L}$, or by another bilinear functional related with $\mathscr {L}$, which allows us to reformulate the orthogonality in a much simpler way in the case of Laguerre polynomials and in a totally explicit form in the case of Jacobi and Gegenbauer polynomials.

A unified study of orthogonal polynomials via Lie algebra

In this paper, we discuss some operators defined on Lie algebras for the purpose of deriving properties of some special functions. The method developed in this paper can also be used to study some other special functions of mathematical physics. We have established a general theorem concerning eigenvectors for the product of two operators defined on a Lie algebra of endomorphisms of a vector space. Further, using this result, we have obtained differential recurrence relations and differential equations for the extended Jacobi polynomials and the Gegenbauer polynomials. Results of many researchers; see for example Radulescu (1991), Mandal (1991), Pathan and Khan (2003), Humi, and the references therein, follow as special cases of our results.

Duffin–Kemmer–Petiau oscillator with Snyder-de Sitter algebra

We present an exact solution of the one-dimensional Bosonic oscillator for spin 1 and spin 0 particles with the Snyder-de Sitter model, where the energy eigenvalues and eigenfunctions are determined for both cases. The wave functions can be given in terms of Gegenbauer polynomials. We also comment on the thermodynamic properties of the system.

Oblique Water Wave Diffraction by a Step

This paper is concerned with the problem of diffraction of an obliquely incident surface water wave train on an obstacle in the form of a finite step. Havelock expansions of water wave potentials are used in the mathematical analysis to obtain the physical parameters reflection and transmission coefficients in terms of integrals. Appropriate multi-term Galerkin approximations involving ultraspherical Gegenbauer polynomials are utilized to obtain a very accurate numerical estimate for reflection and transmission coefficients which are depicted graphically. From these figures various interesting results are discussed.

A unified method for the response analysis of interval/random variable models of acoustic fields with uncertain‐but‐bounded parameters

SummaryFor the response analysis of engineering systems with uncertain‐but‐bounded parameters, three uncertain models have been considered according to the available probability distribution information. One is the bounded random model in which the uncertain parameters are well defined with sufficient probability distribution information and described as bounded random variables. The second one is the interval model in which the uncertain parameters are expressed as interval variables without giving any information of probability distribution. The last one is the bounded hybrid uncertain model, which includes both bounded random variables and interval variables. On the basis of Gegenbauer polynomial approximation theory, a unified interval and random Gegenbauer series expansion (IRGSE) method is proposed and extended for the response prediction of three uncertain models of acoustic fields with uncertain‐but‐bounded parameters. In IRGSE, the uncertain‐but‐bounded variables with different probability distribution information, including interval variables and bounded random variables with different probability density functions, are transformed into the function of unitary variables defined on [‐1,1] associated with the corresponding polynomial parameter (λ) of Gegenbauer series expansion (GSE). The coefficients of GSE is calculated by Gauss–Gegenbauer integration method. By using IRGSE, the responses of three uncertain acoustic models are approximated uniformly by GSE, through which the interval and random analysis can be easily implemented by many numerical solvers. Two numerical examples are applied to investigate the effectiveness of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.

The fractional dimensional spatiotemporal accessible solitons supported by PT-symmetric complex potential

Using the separation variable technique, the properties of localized accessible soliton family supported by a parity-time (PT) symmetric complex potential in fractional dimension (FD) 2<D≤3 are investigated in strongly nonlocal nonlinear media. Analytical solution of the FD Schrödinger equation in the limit of strongly nonlocal nonlinearity, given in terms of the generalized Laguerre and the Gegenbauer polynomials in spherical coordinates, is obtained. Some dynamical characteristics of the accessible solitons and a specific angular expression, such as the spatiotemporal distribution, the angular distribution and external profiles of the analytical solution, are displayed for some peculiar quantum numbers. We expect that the FD accessible solitons will find various applications in science and technology in the future.

Operator Methods and SU(1,1) Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials

Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.

Higher Spin Generalisation of the Gegenbauer Polynomials

In this paper we generalise the harmonic Gegenbauer polynomials to the higher spin setting. To do so we will consider the space of simplicial harmonics and look for polynomials that are invariant with respect to a particular subalgebra of the orthogonal Lie algebra. Analogue to the classic case we will construct a ladder operator which generates our special functions and use them to construct Appell sequences.