Volume 9, pp. 112-127, 1999.

Q-classical orthogonal polynomials: a very classical approach

F. Marcellán and J. C. Medem

Abstract

The $\, q-$classical orthogonal polynomials defined by Hahn satisfy a Sturm-Liouville type equation in geometric differences. Working with this, we classify the $\, q-$classical polynomials in twelve families according to the zeros of the polynomial coefficients of the equation and the behavior concerning to $\, q^{-1} \,$. We determine a $\, q-$analogue of the weight function for the twelve families, and we give a representation of its orthogonality relation and its $\, q-$integral. We describe this representation in some normal and special cases (indeterminate moment problem and finite orthogonal sequences). Finally, the Sturm-Liouville type equation allows us to establish the correspondence between this classification and the Askey Scheme.

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Key words

orthogonal $q-$polynomials, classical polynomials.

AMS subject classifications

33D25.

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