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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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