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

Keywords:: Askey–Wilson class orthogonal polynomials, Askey–Wilson polynomials, Stieltjes–Wigert polynomials, continuous $q^{{-1}}$ -Hermite polynomials, $q$ -Hahn class orthogonal polynomials, $q$ -Laguerre polynomials
Referenced by:: §17.3(v)
Permalink:: http://dlmf.nist.gov/18.29

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

Symbols:: $e$ : base of exponential function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.29.E1
Encodings:: TeX, pMML, png

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.

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\sim$ : asymptotic equality, $z$ : complex variable, $q$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.29.E2
Encodings:: TeX, pMML, png

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).

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

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