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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]

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]

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]

The aim of this paper is to give various properties of homogeneous operators associated with Chan-Chyan-Srivastava polynomials and, by using these results, to obtain harmonic functions by applying Laplace and ultrahyperbolic operators to the Chan-Chyan-Srivastava polynomials. [ABSTRACT FROM AUTHOR]

The main object of this investigation is to define a multivariable matrix generalization of Gould-Hopper polynomials and to reveal some relations such as matrix generating function, matrix recurrence relation, matrix differential equation for them. Furthermore, more general families of bilinear and bilateral matrix generating functions are obtained for these matrix polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we deal with some properties of the matrix extension of the multivariable Humbert polynomials defined by Aktas et.al [Aktas, R., Çekim, B. and Sahin, R., The matrix version for the multivariable Humbert polynomials, Miskolc Mathematical Notes, 13(2) (2012), 197-208]. We give differential equations for the products of these matrix polynomials and some other multivariable matrix polynomials, and also we present some new relations for the multivariable Humbert matrix polynomials. [ABSTRACT FROM AUTHOR]

We introduce a Kantorovich-Stancu type modification of a generalization of Szasz operators defined by means of the Brenke type polynomials and obtain approximation properties of these operators. Also, we give a Voronovskaya type theorem for Kantorovich- Stancu type operators including Gould-Hopper polynomials. [ABSTRACT FROM AUTHOR]

In this paper, the matrix extension of the multivariable Humbert polynomials is introduced. Various families of linear, multilinear and multilateral generating matrix functions of these matrix polynomials are presented. Miscellaneous applications are also discussed. [ABSTRACT FROM AUTHOR]

We give a Kantorovich variant of a generalization of Szasz operators defined by means of the Brenke-type polynomials and obtain convergence properties of these operators by using Korovkin's theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity, and Peetre's K-functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronov-skaya-type result is given for these operators including Gould-Hopper polynomials. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a multivariable extension of the Humbert polynomials, which is motivated by the Chan–Chyan–Srivastava multivariable polynomials, the multivariable extension of the familiar Lagrange–Hermite polynomials and Erkus–Srivastava multivariable polynomials. We derive various families of multilinear and mixed multilateral generating functions for these polynomials. Other miscellaneous properties of these multivariable polynomials are also discussed. Some special cases of the results presented in this study are also indicated. [Copyright &y& Elsevier]