The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function Wγ (x) = x γ 1 1 · · · x γ d d (1 − |x|) γ d+1 when all γ i > −1 and they are eigenfunctions of a second order partial differential operator Lγ . The singular cases that some, or all, γ 1 , . . . , γ d+1 are −1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of Lγ in each singular case is found. Secondly, these polynomials are shown to be orthogonal with respect to an inner product which is explicitly determined. This inner product involves derivatives of the functions, hence the name Sobolev orthogonal polynomials.

The aim of the present paper is to introduce Dunkl-Gamma type operators in terms of Appell polynomials and to investigate approximating properties of these operators.2000 Mathematics Subject Classification. Primary 41A25, 41A36; Secondary 33C45.

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.

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 -functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronovskaya-type result is given for these operators including Gould-Hopper polynomials.

ABSTRACT. In this paper, we give some relations between multivariable Laguerre polynomials and other well-known multivariable polynomials. We get various families of multilinear and multilateral generating functions for these polynomials. Some special cases are also presented.