Mehler–Heine formula

In mathematics, the Mehler–Heine formula introduced by Gustav Ferdinand Mehler and Eduard Heine describes the asymptotic behavior of the Legendre polynomials as the index tends to infinity, near the edges of the support of the weight. There are generalizations to other classical orthogonal polynomials, which are also called the Mehler–Heine formula. The formula complements the Darboux formulae which describe the asymptotics in the interior and outside the support.

Legendre polynomials

The simplest case of the Mehler–Heine formula states that :\lim _{n\to\infty}P_n\left(\cos{\frac{z}{n}}\right) = \lim _{n\to\infty}P_n\left(1-\frac{z^2}{2n^2}\right) = J_0(z), where is the Legendre polynomial of order , and the Bessel function of order 0. The limit is uniform over in an arbitrary bounded domain in the complex plane. ==Jacobi polynomials==

Jacobi polynomials

The generalization to Jacobi polynomials is given by Gábor Szegő as follows :\lim_{n \to \infty} n^{-\alpha}P_n^{(\alpha,\beta)}\left(\cos \frac{z}{n}\right) = \lim_{n \to \infty} n^{-\alpha}P_n^{(\alpha,\beta)}\left(1-\frac{z^2}{2n^2}\right) = \left(\frac{z}{2}\right)^{-\alpha} J_\alpha(z), where is the Bessel function of order. ==Laguerre polynomials==

Laguerre polynomials

Using generalized Laguerre polynomials and confluent hypergeometric functions, they can be written as :\lim_{n \to \infty} n^{-\alpha}L_n^{(\alpha)}\left(\frac{z^2}{4n}\right) = \left(\frac{z}{2}\right)^{-\alpha} J_\alpha(z), where is the Laguerre function. ==Hermite polynomials==

Hermite polynomials

Using the expressions relating Hermite polynomials and Laguerre polynomials where two equations exist, they can be written as :\begin{align}\lim_{n \to \infty} \frac{(-1)^n}{4^nn!}\sqrt{n}H_{2n}\left(\frac{z}{2\sqrt{n}}\right) &=\left(\frac{z}{2}\right)^{\frac{1}{2}}J_{-\frac{1}{2}}(z) \\ \lim_{n \to \infty} \frac{(-1)^n}{4^nn!}H_{2n+1}\left(\frac{z}{2\sqrt{n}}\right) &=(2z)^{\frac{1}{2}}J_{\frac{1}{2}}(z),\end{align} where is the Hermite function. ==References==

Source: Wikipedia ↗

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