A Mehler-Heine formula for disk polynomials
Let \(R_ n^{(\alpha,\beta)}\) be the Jacobi polynomials normalized by \(R_ n^{(\alpha,\beta)}(1)=1\). The classical Mehler-Heine formula compares the \(R_ n^{(\alpha,\beta)}\) with the Bessel functions \(J_{\alpha}\) of the first kind. This formula has been used in \textit{L. Gallardo} [Adv. Appl. Probab. 16, 293-323 (1984; Zbl 0542.60071)] and, in a more general way, in \textit{Voit} [ibid. 22, No.1, 68-87 (1990); Zbl 0719.60009)] to derive central limit theorems for random walks on the nonnegative integers whenever the transition probabilities are given in terms of the linearization coefficients of the polynomials \(R_ n^{(\alpha,\beta)}\). The limit has been a Rayleigh distribution. This fundamental paper deals with the two-dimensional situation. Let \(R^{\alpha}_{m,n}\) be the so-called disk polynomials which are defined by \[ R^{\alpha}_{m,n}(\rho e^{i\theta})=e^{i(m-n)\theta} \rho^{| m-n|}R_{m\wedge n}^{(\alpha,| m- n|)}(2\rho^ 2-1). \] These polynomials are orthogonal with respect to a measure supported by the unit ball in \({\mathbb{R}}^ 2\). If \(\alpha =d- 2>0\), \(d\in {\mathbb{N}}\), then the \(R^{\alpha}_{m,n}\) may be interpreted as the zonal spherical functions of the Gelfand pair (U(d),U(d-1)). As a main result it is shown that \[ \lim_{n,m\to \infty}R^{\alpha}_{m,n}(\cos \frac{x}{2\sqrt{mn}})=2^{\alpha}\Gamma (\alpha +1)x^{-\alpha}J_{\alpha}(x) \] uniformly on compact intervals where bounds for the rate of convergence of this limit are given. This two-dimensional version of the Mehler-Heine formula is applied to Markov chains on \({\mathbb{N}}_ 0\times {\mathbb{N}}_ 0\) where the transition probabilities are associated with the polynomials \(R^{\alpha}_{m,n}\). This leads to a new type of a central limit theorem where the limit distribution, a two-dimensional analog of the Rayleigh distribution, is supported by \([0,\infty [^ 2\) and has the Lebesgue density \[ \frac{2^{\alpha +1}}{\sqrt{2\pi}\Gamma (\alpha +1)}(xy)^{\alpha}(x+y)e^{-(x+y)^ 2/2}\quad (x,y\geq 0). \]
scientific article; zbMATH DE number 4200602