Mehler–Heine formula

Remarks

Proof idea: write $P_n(\cos \theta) = {}_2F_1\left( -n, n + 1; 1; \sin^2 \frac{\theta}{2} \right) = \sum_{k=0}^{n} \frac{(-n)_k (n + 1)_k}{(k!)^2} \sin^{2k} \frac{\theta}{2}$. With $\theta = z/n$ and $k$ fixed, $(-n)_k (n + 1)_k \sim (-1)^k n^{2k}$ and $\sin^{2k} \frac{z}{2n} \sim \left( \frac{z}{2n} \right)^{2k}$. So the $k$-th term tends to $\frac{(-1)^k}{(k!)^2} \left( \frac{z}{2} \right)^{2k}$, the $k$-th term of the series of $J_0(z)$, and a domination argument allows the termwise limit. The same holds for Jacobi polynomials: $\lim_{n \to \infty} n^{-\alpha} P_n^{(\alpha, \beta)}\left( \cos \frac{z}{n} \right) = \left( \frac{z}{2} \right)^{-\alpha} J_\alpha(z)$. A consequence via Hurwitz's theorem: if $\theta_{n,k}$ is the $k$-th smallest zero of $\theta \mapsto P_n(\cos \theta)$ in $(0, \pi)$, then $n \theta_{n,k} \to j_{0,k}$, the $k$-th positive zero of $J_0$.

Going deeper

For a student

The formula describes the behaviour of orthogonal polynomials at the edge of their interval, where the oscillatory asymptotics in the interior break down. In the interior, the classical asymptotic formulas give cosine waves with slowly varying amplitude; near $x = \pm 1$ Bessel functions take their place. It explains why the zeros of $P_n$ closest to $1$, measured in the angle $\theta$, are about $j_{0,k}/n$. Its versions for Jacobi and Laguerre polynomials, and their analogues for random matrices, describe the "hard edge" behaviour of eigenvalues.