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

Ismail (1986) gives asymptotic expansions as $n\to \mathrm{\infty }$, 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 $p_n(\cos \theta ;a,b,c,d|q)$ the leading term is given by

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

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).