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Asymptotics of $$q$$-orthogonal polynomials and a $$q$$-Airy function. (English) Zbl 1072.33014
Let $$0<q<1$$. The polynomials $$h_{n}(x\,| \,q)$$ satisfy the recurrence relation $2xh_{n}(x\,| \,q)=h_{n+1}(x\,| \,q)+q^{-n}(1-q^n)h_{n-1}(x\,| \,q)$ with $$h_{0}(x\,| \,q)=1$$ and $$h_{1}(x\,| \,q)=2x$$. These polynomials are $$q$$-analogues of the Hermite polynomials $$H_{n}(x)$$. A $$q$$-analogue of the Laguerre polynomials $$L_{n}^{(\alpha)}(x)$$ is the system of polynomials $$L_{n}^{(\alpha)}(x;q)$$ generated by
$\begin{split} -xq^{2n+\alpha+1}L_{n}^{(\alpha)}(x;q)=\\ (1-q^{n+1})L_{n+1}^{(\alpha)}(x;q)+ q(1-q^{n+\alpha})L_{n-1}^{(\alpha)}(x;q)] -[(1-q^{n+1})+ q(1-q^{n+\alpha})]L_{n}^{(\alpha)}(x;q) \end{split}$ and the initial conditions
$L_{0}^{(\alpha)}(x;q)=1,\quad L_{1}^{(\alpha)}(x;q)= \frac{1-q^{\alpha+1}- xq^{\alpha+1}}{1-q}.$ In this paper, the author establishes complete asymptotic expansions of $$h_{n}(x\,| \,q)$$ and $$L_{n}^{(\alpha)}(x;q)$$ around their largest zero. A similar formula is also derived for the Stieltjes-Wigert polynomials $S_{n}(x;q)= \sum_{k=0}^{n}\frac{q^{k^2}(-x)^{k}}{(q;q)_{k}(q;q)_{n-k}}.$ In doing so, he discovers a $$q$$-analogue of the Airy function, namely $A_{q}(z)=\sum_{k=0}^{\infty}\frac{q^{k^2}(-z)^{k}}{(q;q)_{k}}.$ He also derives Plancherel-Rotach-type asymptotics for the polynomials $$Q_{n}(x)$$ generated by $Q_{0}(x)=1,\;Q_{1}(x)=2x-a-b,\;Q_{n+1}(x)= [2x-(a+b)q^{-n}]Q_{n}(x)-(ab+q^{n-1})Q_{n-1}(x).$ The term Plancherel-Rotach asymptotics refers to asymptotics around the largest and smallest zeros. The polynomials $$Q_{n}(x)$$ are Al-Salam-Chichara polynomials with $$q \rightarrow 1/q$$. The author also establishes asymptotic formulas of Plancherel-Rotach-type for all $$q$$-polynomials in the Askey scheme, which are orthogonal on unbounded sets.

MSC:
 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$ 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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