Orthogonal Polynomial Solutions Research Articles

On a Symmetric Generalization of Bivariate Sturm–Liouville Problems

A new class of partial differential equations having symmetric orthogonal solutions is presented. The general equation is presented and orthogonality is obtained using the Sturm–Liouville approach. Conditions on the polynomial coefficients to have admissible partial differential equations are given. The general case is analyzed in detail, providing orthogonality weight function, three-term recurrence relations for the monic orthogonal polynomial solutions, as well as explicit form of these monic orthogonal polynomial solutions, which are solutions of an admissible and potentially self-adjoint linear second-order partial differential equation of hypergeometric type.

Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or a q-quadratic lattice

ABSTRACTIf is an orthogonal polynomial system, then satisfies a three-term recurrence relation of type with . On the other hand, Favard's theorem states that the converse is true. A general method to derive the coefficients , , in terms of the polynomial coefficients of the divided-difference equations satisfied by orthogonal polynomials on a quadratic or q-quadratic lattice is recalled. If a three-term recurrence relation is given as input, the Maple implementations rec2ortho of Koorwinder and Swarttouw or retode of Koepf and Schmersau can identify its solution which is a (linear transformation of a) classical orthogonal polynomial system of a continuous, a discrete or a q-discrete variable, if applicable. The two implementations rec2ortho and retode do not handle classical orthogonal polynomials on a quadratic or q-quadratic lattice. Motivated by an open problem, submitted by Alhaidari during the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications, which will serve as application, the Maple implementation retode of Koepf and Schmersau is extended to cover classical orthogonal polynomial solutions on quadratic or q-quadratic lattices of three-term recurrence relations.

Comparison of the Orthogonal Polynomial Solutions for Fractional Integral Equations

In this paper, a collocation method based on the orthogonal polynomials is presented to solve the fractional integral equations. Six numerical examples are given to illustrate the method. The results are compared with the other methods in the literature, and the results obtained by different kinds of polynomials are compared.

On Linear Ordinary Differential Equation With Orthogonal Polynomial Solutions

: The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. In this paper we give a survey of the orthogonal polynomial solutions of second-order and fourth-order linear ordinary differential equation, on the generated self-adjoint differential operators .

Two finite q-Sturm-Liouville problems and their orthogonal polynomial solutions

In this paper, we consider two new q-Sturm-Liouville problems and prove that their polynomial solutions are finitely orthogonal with respect to two weight functions which correspond to Fisher and Tstudent distributions as q ? 1. Then, we obtain the general properties of these polynomial solutions, such as orthogonality relations, three term recurrence relations, q-difference equations and basic hypergeometric representations, where all results in the continuous case are recovered as q ? 1.

Interbasis expansions in the Zernike system

The differential equation with free boundary conditions on the unit disk that was proposed by Frits Zernike in 1934 to find Jacobi polynomial solutions (indicated as I) serves to define a classical system and a quantum system which have been found to be superintegrable. We have determined two new orthogonal polynomial solutions (indicated as II and III) that are separable and involve Legendre and Gegenbauer polynomials. Here we report on their three interbasis expansion coefficients: between the I–II and I–III bases, they are given by F23(⋯|1) polynomials that are also special su(2) Clebsch–Gordan coefficients and Hahn polynomials. Between the II–III bases, we find an expansion expressed by F34(⋯|1)’s and Racah polynomials that are related to the Wigner 6j coefficients.

Linear partial q-difference equations on q-linear lattices and their bivariate q-orthogonal polynomial solutions

Orthogonal polynomial solutions of an admissible potentially self-adjoint linear second-order partial q-difference equation of the hypergeometric type in two variables on q-linear lattices are analyzed. A q-Pearson’s system for the orthogonality weight function, as well as for the difference derivatives of the solutions are presented, giving rise to a solution of the q-difference equation under study in terms of a Rodrigues-type formula. The monic orthogonal polynomial solutions are treated in detail, giving explicit formulae for the matrices in the corresponding recurrence relations they satisfy. Lewanowicz and Woźny [S. Lewanowicz, P. Woźny, J. Comput. Appl. Math. 233 (2010) 1554–1561] have recently introduced a (non-monic) bivariate extension of big q-Jacobi polynomials together with a partial q-difference equation of the hypergeometric type that governs them. This equation is analyzed in the last section: we provide two more orthogonal polynomial solutions, namely, a second non-monic solution from the Rodrigues’ representation, and the monic solution both from the recurrence relation that govern them and also explicitly given in terms of generalized bivariate basic hypergeometric series. Limit relations as q↑1 for the partial q-difference equation and for the all three q-orthogonal polynomial solutions are also presented.

On a class of bivariate second-order linear partial difference equations and their monic orthogonal polynomial solutions

In this paper we classify the bivariate second-order linear partial difference equations, which are admissible, potentially self-adjoint, and of hypergeometric type. Using vector matrix notation, explicit expressions for the coefficients of the three-term recurrence relations satisfied by monic orthogonal polynomial solutions are obtained in terms of the coefficients of the partial difference equation. Finally, we make a compilation of the examples existing in the literature belonging to the class analyzed in this paper, namely bivariate Charlier, Meixner, Kravchuk and Hahn orthogonal polynomials.

Bivariate second-order linear partial differential equations and orthogonal polynomial solutions

In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second-order linear partial differential equations, which are admissible potentially self-adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation, using vector matrix notation. Moreover, Rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given. As illustration, these results are applied to a two parameter monic Appell polynomials. Finally, the non-monic case is briefly discussed.

Linear partial difference equations of hypergeometric type: Orthogonal polynomial solutions in two discrete variables

In this paper a systematic study of the orthogonal polynomial solutions of a second order partial difference equation of hypergeometric type of two variables is done. The Pearson's systems for the orthogonality weight of the solutions and also for the difference derivatives of the solutions are presented. The orthogonality property in subspaces is treated in detail, which leads to an analog of the Rodrigues-type formula for orthogonal polynomials of two discrete variables. A classification of the admissible equations as well as some examples related with bivariate Hahn, Kravchuk, Meixner, and Charlier families, and their algebraic and difference properties are explicitly given.

A construction of real weight functions for certain orthogonal polynomials in two variables

H.L. Krall and I.M. Sheffer considered the problem of classifying certain second-order partial differential equations having an algebraically complete, weak orthogonal bivariate polynomial system of solutions. Two of the equations that they considered are ( x 2 + y ) u x x + 2 x y u x y + y 2 u y y + g x u x + g ( y − 1 ) u y = λ u , and x 2 u x x + 2 x y u x y + ( y 2 − y ) u y y + g ( x − 1 ) u x + g ( y − γ ) u y = λ u . Even though they showed that these equations have a sequence of weak orthogonal polynomial solutions, they were unable to show that these polynomials were, in fact, orthogonal. The orthogonality of these two polynomial sequences was recently established by Kwon, Littlejohn, and Lee solving an open problem from 1967. In this paper, we construct explicit weight functions for these two orthogonal polynomial sequences, using a method first developed by Littlejohn and then further developed by Han, Kim, and Kwon. Moreover, two additional partial differential equations were found by Kwon, Littlejohn, and Lee that have sequences of orthogonal polynomial solutions. These equations are given by ( x 2 − x ) u x x + 2 x y u x y + y 2 u y y + ( d x + e ) u x + ( d y + h ) u y = λ u , x u x x + 2 y u x y + ( d x + e ) u x + ( d y + h ) u y = λ u . In each of these examples, we also produce explicit orthogonalizing weight functions.

Recurrence equations and their classical orthogonal polynomial solutions

The classical orthogonal polynomials are given as the polynomial solutions p n ( x) of the differential equation σ(x)y″(x)+τ(x)y ′(x)+λ ny(x)=0, where σ( x) is a polynomial of at most second degree and τ( x) is a polynomial of first degree. In this paper a general method to express the coefficients A n, B n and C n of the recurrence equation p n+1(x)=(A nx+B n)p n(x)−C np n−1(x) in terms of the given polynomials σ( x) and τ( x) is used to present an algorithm to determine the classical orthogonal polynomial solutions of any given holonomic three-term recurrence equation, i.e., a homogeneous linear three-term recurrence equation with polynomial coefficients. In a similar way, classical discrete orthogonal polynomial solutions of holonomic three-term recurrence equations can be determined by considering their corresponding difference equation σ(x)Δ∇y(x)+τ(x)Δy(x)+λ ny(x)=0, where Δy( x)= y( x+1)− y( x) and ∇ y( x)= y( x)− y( x−1) denote the forward and backward difference operators, respectively, and a similar approach applies to classical q-orthogonal polynomials, being solutions of the q-difference equation σ(x)D qD 1/qy(x)+τ(x)D qy(x)+λ q,ny(x)=0, where D qf(x)= f(qx)−f(x) (q−1)x , q≠1, denotes the q-difference operator.

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Qn(x)}∞n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Qn(x)}∞n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: LN[y](x)=∑Ni=1ℓi(x)y(i)(x)=λny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Qn(x)}∞n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.

Orthogonal polynomial solutions of linear ordinary differential equations

This paper surveys the latest known results concerning the classification of differential equations of the form ∑ k=1 N a k(x)y (k)(x)=λy(x) having a sequence of polynomial eigenfunctions { p n ( x)} n=0 ∞ that are orthogonal with respect to some real bilinear form. Since the publishing of the Erice Report in 1990, several new significant results and generalizations have been discovered.

Forced vertical vibrations of an elastic elliptic plate on an elastic half space – a direct approach using orthogonal polynomials

The forced vibration of an elastic half space produced by a rigid elliptic indenter oscillating about an axis perpendicular to the plane face of the half space is considered. The boundary conditions lead to a two-dimensional dual integral equation in terms of the unknown normal stress. By appropriate substitution, the dual integral equation is first reduced to a two-dimensional Fredholm integral equation. This is transformed to an infinite set of equations using Abelian transformations. Next, the Abel-transformed variable of the unknown normal stress is expanded in terms of orthogonal Jacobi polynomials, and by solving the system of linear equations, orthogonal polynomial solutions are obtained. The method used to obtain the orthogonal polynomial solutions of this problem is new and the major advantage of this expansion technique is that it is valid for all frequencies. Detailed numerical work has been given for the total load on the disc for different values of frequencies.

Partial Differential Equations and Bivariate Orthogonal Polynomials

In 1929, S. Bochner identified the families of polynomials which are eigenfunctions of a second-order linear differential operator. What is the appropriate generalization of this result to bivariate polynomials? One approach, due to Krall and Sheffer in 1967 and pursued by others, is to determine which linear partial differential operators have orthogonal polynomial solutions with all the polynomials in the family of the same degree sharing the same eigenvalue. In fact, such an operator only determines a multi-dimensional eigenspace associated with each eigenvalue; it does not determine the individual polynomials, even up to a multiplicative constant. In contrast, our approach is to seek pairs of linear differential operators which have joint eigenfunctions that comprise a family of bivariate orthogonal polynomials. This approach entails the addition of some “normalizing" or “regularity" conditions which allow determination of a unique family of orthogonal polynomials. In this article we formulate and solve such a problem and show with the help of Mathematica that the only solutions are disk polynomials. Applications are given to product formulas and hypergroup measure algebras.

Partial differential equations having orthogonal polynomial solutions

We show that if a second order partial differential equation: L[ u]:= Au xx + 2 Bu xy + Cu yy + Du x + Eu y = λ n u has orthogonal polynomial solutions, then the differential operator L[·] must be symmetrizable and can not be parabolic in any nonempty open subset of the plane. We also find Rodrigues type formula for orthogonal polynomial solutions of such differential equations.

Symmetrizability of differential equations having orthogonal polynomial solutions

We show that if a linear differential equation of spectral type with polynomial coefficients L N[y](x) = ∑ i=0 N l i(x) = λ ny(x) has an orthogonal polynomial system of solutions, then the differential operator L N [·] must be symmetrizable. We also give a few applications of this result.

Differential equations having orthogonal polynomial solutions

Necessary and sufficient conditions for an orthogonal polynomial system (OPS) to satisfy a differential equation with polynomial coefficients of the form (∗) L N[y] = ∑ i=1 N l i(x)y (i)(x) = λ ny(x) were found by H.L. Krall. Here, we find new necessary conditions for the equation (∗) to have an OPS of solutions as well as some other interesting applications. In particular, we obtain necessary and sufficient conditions for a distribution w( x) to be an orthogonalizing weight for such an OPS and investigate the structure of w( x). We also show that if the equation (∗) has an OPS of solutions, which is orthogonal relative to a distribution w( x), then the differential operator L N [·] in (∗) must be symmetrizable under certain conditions on w( x).

Recurrence relations for connection coefficients between two families of orthogonal polynomials

We describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second- and fourth-order differential equations.