18 Orthogonal Polynomials Other Orthogonal Polynomials 18.28 Askey–Wilson Class 18.30 Associated OP’s

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

ⓘ

Keywords:: Askey–Wilson class orthogonal polynomials, Askey–Wilson polynomials, Stieltjes–Wigert polynomials, asymptotic approximations, asymptotic approximations to zeros, 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
See also:: Annotations for Ch.18

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 $p_{n}\left(\cos\theta;a,b,c,d\,|\,q\right)$ the leading term is given by

18.29.1		$\left(bc,bd,cd;q\right)_{n}\*\left(Q_{n}({\mathrm{e}}^{\mathrm{i}\theta};a,b,c% ,d\mid q)+Q_{n}({\mathrm{e}}^{-\mathrm{i}\theta};a,b,c,d\mid q)\right),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/18.29.E1 Encodings: TeX, pMML, png See also: Annotations for §18.29 and Ch.18

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

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: $\sim$ : asymptotic equality, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $z$ : complex variable, $q$ : real variable and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/18.29.E2 Encodings: TeX, pMML, png See also: Annotations for §18.29 and Ch.18

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).

18.28 Askey–Wilson Class 18.30 Associated OP’s

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

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