Mehler–Heine formula

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 P**n is the Legendre polynomial of order n, and J0 the Bessel function of order 0. The limit is uniform over z in an arbitrary bounded domain in the complex plane.

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 J*α* is the Bessel function of order α.

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

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 H**n is the Hermite function.

References

References

1. Mehler, G.F.. (1868). "Ueber die Vertheilung der statischen Elektricität in einem von zwei Kugelkalotten begrenzten Körper". Journal für die Reine und Angewandte Mathematik.
2. Heine, E.. (1861). "Handbuch der Kugelfunktionen. Theorie und Anwendung.". Georg Reimer.
3. Szegő, Gábor. (1939). "Orthogonal Polynomials". American Mathematical Society.
4. Koekoek, Roelof. (2010). "Hypergeometric Orthogonal Polynomials and Their q-Analogues". Springer-Verlag.