
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 \[ \Theta_q^{+}(z):= \sum_{k=0}^{\infty} q^{k^2} z^k. \]
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
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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