# §18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

Ismail (1986) gives asymptotic expansions as $n\to\infty$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. These asymptotic expansions are in fact convergent expansions. For Askey–Wilson $\mathop{p_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta;a,b,c,d\,|\,q\right)$ the leading term is given by

 18.29.1 $\left(bc,bd,cd;q\right)_{n}\*\left(Q_{n}(e^{i\theta};a,b,c,d\mid q)+Q_{n}(e^{-% i\theta};a,b,c,d\mid q)\right),$

where with $z=e^{\pm i\theta}$,

 18.29.2 $Q_{n}(z;a,b,c,d\mid q)\sim\frac{z^{n}\left(az^{-1},bz^{-1},cz^{-1},dz^{-1};q% \right)_{\infty}}{\left(z^{-2},bc,bd,cd;q\right)_{\infty}},$ $n\to\infty$; $z,a,b,c,d,q$ fixed.

For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006).

For asymptotic approximations to the largest zeros of the $q$-Laguerre and continuous $q^{-1}$-Hermite polynomials see Chen and Ismail (1998).