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

Ismail (1986) gives asymptotic expansions as n, 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 pn(cosθ;a,b,c,d| q) the leading term is given by

18.29.1 (bc,bd,cd;q)n(Qn(θ;a,b,c,dq)+Qn(-θ;a,b,c,dq)),

where with z=±θ,

18.29.2 Qn(z;a,b,c,dq)zn(az-1,bz-1,cz-1,dz-1;q)(z-2,bc,bd,cd;q),
n; 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).