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§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

where with z=e^{{\pm 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).