## 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.

Full Text (PDF) [161 KB]

### Key words

orthogonal $q-$polynomials, classical polynomials.

33D25.

< Back