Indeterminate Cases within the Askey-Wilson Scheme

Abstract

The orthogonal polynomials associated with an indeterminate moment problem have infinitely many measures of orthogonality. The one parameter family of $N$-extremal measures $\sigma(x,\lambda)$ are discribed in terms of four entire functions $A(z)$, $B(z)$, $C(z)$ and $D(z)$ by $$\int \frac{d\sigma(x,\lambda)}{z-x}=(A(z)-\lambda C(z))/ (B(z)-\lambda D(z)), \qquad \lambda \in R.$$
The Askey-Wilson polynomials are described in terms of four parameters $a$, $b$, $c$, $d$. We take a limit when one parameter, say $d$, tends to infinity to obtain a new family of orthogonal polynomials associated with an indeterminate case. We show how to obtain explicit expression for the corresponding entire functions $A(z)$, $B(z)$, $C(z)$ , $D(z)$ in terms of $q$-hypergeometric functions.