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Uniform asymptotics of some q-orthogonal polynomials

Uniform asymptotics of some \(q\)-orthogonal polynomials
Authors: Wang, X.S.; Wong, R.;

Uniform asymptotics of some q-orthogonal polynomials

Abstract

The authors derive uniform asymptotic formulas for the Stieltjes-Wigert polynomial, given by \[ s_n(z;q)=\sum_{k=0}^{n} {{q^{k^2}}\over{(q;q)_k (q;q)_{n-k}}} (-z)^k, \] the \(q^{-1}\)-Hermite polynomial and the \(q\)-Laguerre polynomial, as the degree of the polynomials tends to infinity. These formulas involve what the authors call the \(q\)-Airy polynomial, defined by \[ A_{q,n}(z) := \sum_{k=0}^{n} {q^{k^2} \over (q;q)_k} (-z)^k, \] and the half \(q\)-Theta function, defined by \[ \Theta_q^{+}(z):= \sum_{k=0}^{\infty} q^{k^2} z^k. \]

Related Organizations
Keywords

q−1-Hermite polynomial, Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.), Asymptotic approximations, asymptotic expansions (steepest descent, etc.), Stieltjes-Wigert polynomial, Applied Mathematics, q-Theta function, \(q\)-Laguerre polynomial, Asymptotic representations in the complex plane, Stieltjes–Wigert polynomial, q-Laguerre polynomial, q-Airy function, \(q^{-1}\)-Hermite polynomial, \(q\)-theta function, Analysis, \(q\)-Airy function

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
3
Average
Average
Average
hybrid