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In this paper, Fourier transform of multivariate orthogonal polynomials on the simplex is presented. A new family of multivariate orthogonal functions is obtained using the Parseval's identity and several recurrence relations are derived. [ABSTRACT FROM AUTHOR]

Some systems of univariate orthogonal polynomials can be mapped into other families by the Fourier transform. The most-studied example is related to the Hermite functions, which are eigenfunctions of the Fourier transform. For the multivariate case, by using the Fourier transform and Parseval's identity, very recently, some examples of orthogonal systems of this type have been introduced and orthogonality relations have been discussed. In the present paper, this method is applied for multivariate orthogonal polynomials on the unit ball. The Fourier transform of these orthogonal polynomials on the unit ball is obtained. By Parseval's identity, a new family of multivariate orthogonal functions is introduced. The results are expressed in terms of the continuous Hahn polynomials. [ABSTRACT FROM AUTHOR]

Three term relations for orthogonal polynomials in several variables associated to a moment linear functional obtained by a Uvarov modification of a given moment functional are studied. Existence of Uvarov orthogonal polynomials is analyzed, stating conditions to ensure it. The matrices of the three term relations of the Uvarov orthogonal polynomials are explicitly given in terms of the matrices of the three term relations satisfied by the original family. Two examples are presented in order to show that the results are valid for positive definite linear functionals and also for some quasi definite linear functionals which are not positive definite. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of paper is to give some limit relationships between finite and infinite sequences of orthogonal polynomials in two variables and to obtain the well-known relations of some infinite sets of two-variable orthogonal polynomials by taking limit of the properties verified by finite classes of two-variable orthogonal polynomials. Furthermore, the fourth-order partial differential equations satisfied by the polynomials, which are the products of finite orthogonal polynomials, are presented and by taking limit of the derived fourth-order equations, partial differential equations for some infinite sequences of two-variable orthogonal polynomials are found. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce several finite sets of orthogonal polynomials in two variables, and investigate some general properties of them such as recurrence relations, generating functions, differential equations, and Rodrigues type representations. [ABSTRACT FROM AUTHOR]

In this paper we introduce a six-parameter generalization of the four-parameter three-variable polynomials on the simplex and we investigate the properties of these polynomials. Sparse recurrence relations are derived by using ladder relations for shifted univariate Jacobi polynomials and bivariate polynomials on the triangle. Via these sparse recurrence relations, second order partial differential equations are presented. Some connection relations are obtained between these polynomials. Also, new results for the four-parameter three-variable polynomials on the simplex are given. Finally, some generating functions are derived. [ABSTRACT FROM AUTHOR]

In this paper some new classes of two-variable orthogonal functions by using Fourier transforms of two-variable orthogonal polynomials are introduced. Orthogonality relations are obtained by using the Parseval identity. Recurrence relations for new families of orthogonal functions are also presented. [ABSTRACT FROM AUTHOR]

The present paper deals with various recurrence relations, generating functions and series expansion formulas for two families of orthogonal polynomials in two variables, given Laguerre–Laguerre Koornwinder polynomials and Laguerre–Jacobi Koornwinder polynomials in the limit cases. Several families of bilinear and bilateral generating functions are derived. Furthermore, some special cases of the results presented in this study are indicated. [ABSTRACT FROM AUTHOR]

In this paper, we give the parameter derivative representations in the form of ∂ P n , k ( λ ; x , y ) ∂ λ = ∑ m = 0 n − 1 ∑ j = 0 m d n , j , m P m , j ( λ ; x , y ) + ∑ j = 0 k e n , j , k P n , j ( λ ; x , y ) for some Koornwinder polynomials where λ is a parameter and 0 ≤ k ≤ n ; n = 0 , 1 , 2 , … and present orthogonality properties of the parametric derivatives of these polynomials. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to give the parameter derivative representations of the form ∂ P n , k ( λ ; x , y ) ∂ λ = ∑ m = 0 n - 1 ∑ j = 0 m d n , j , m P m , j ( λ ; x , y ) + ∑ j = 0 k e n , j , k P n , j ( λ ; x , y ) for a family of orthogonal polynomials of variables x and y , with λ being a parameter and 0 ⩽ k ⩽ n ; n , k = 0 , 1 , 2 , … . First, we shall present the representations of the parameter derivatives of the generalized Gegenbauer polynomials C n ( λ , μ ) ( x ) with the help of the parameter derivatives of the classical Jacobi polynomials P n ( α , β ) ( x ) , i.e. ∂ ∂ α P n ( α , β ) ( x ) and ∂ ∂ β P n ( α , β ) ( x ) . Then, by using these derivatives, we investigate the parameter derivatives for two-variable analogues of the generalized Gegenbauer polynomials. Furthermore, we discuss orthogonality properties of the parametric derivatives of these polynomials. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to obtain derivatives of the Jacobi polynomials with two variables on the triangle with respect to their parameters by using the parameter derivatives of the classical Jacobi polynomials P n ( α , β ) ( x ) . Furthermore, we give orthogonality properties of the parametric derivatives on the triangle. [ABSTRACT FROM AUTHOR]

Abstract: The main object of this paper is to construct a two-variable analogue of Jacobi polynomials and to give some properties of these polynomials. We show that these polynomials are orthogonal, then we obtain various recurrence formulas for them. Furthermore, we give some integral representations for these polynomials. [Copyright &y& Elsevier]

Abstract: In this paper, we present various integral representations for the Chan–Chyan–Srivastava, Lagrange–Hermite, Erkus–Srivastava multivariable polynomials and extended Jacobi polynomials. Then, we obtain some partial differential equations for the product of any two of them. We show that the Chan–Chyan–Srivastava multivariable polynomials are not orthogonal. We also discuss other miscellaneous properties of the Chan–Chyan–Srivastava, Lagrange–Hermite and Erkus–Srivastava multivariable polynomials. [Copyright &y& Elsevier]