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

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.)
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