Extensions of discrete classical orthogonal polynomials beyond the orthogonality. (English) Zbl 1167.42008

Costas-Santos, R. S.; Sánchez-Lara, J. F.

J. Comput. Appl. Math. 225, No. 2, 440-451 (2009).

The authors extend definition of the discrete classical orthogonal polynomials to the case of all possible degrees and prove that main properties of the Krawtchouk, Hahn, dual Hahn and Rocah polynomials hold for new (extended) families except the orthogonality. Moreover they find a \(\Delta\)-Sobolev orthogonality of the form \(\langle f,g\rangle_S=\langle{\mathbf u}_0,fg\rangle+\langle {\mathbf u}_1,(\Delta^Mf)(\Delta^Mg)\rangle\) for them, where \(\mathbf u_0\) and \(\mathbf u_1\) are certain classical functionals.

Reviewer: Alexei Lukashov (Istanbul)

MSC:

42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

discrete classical orthogonal polynomials; Sobolev orthogonality

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