Mehler–Heine formula

In mathematics, the Mehler–Heine formula introduced by Mehler (1868) and Heine (1861) 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

${\displaystyle \lim _{n\to \infty }P_{n}{\Bigl (}\cos {z \over n}{\Bigr )}=J_{0}(z)}$

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

Jacobi polynomials

The generalization to Jacobi polynomials P^α,β
_n is given by (Szegő 1939, 8.1) as follows:

${\displaystyle \lim _{n\to \infty }n^{-\alpha }P_{n}^{\alpha ,\beta }\left(\cos {\frac {z}{n}}\right)=\left({\frac {z}{2}}\right)^{-\alpha }J_{\alpha }(z)~.}$

References

• Heine, E. (1861), Handbuch der Kugelfunktionen. Theorie und Anwendung. Neudruck. (in German), Georg Reimer, Berlin, Zbl 0103.29304
• Mehler, F. G. (1868), "Ueber die Vertheilung der statischen Elektricität in einem von zwei Kugelkalotten begrenzten Körper." (PDF), Journal für die Reine und Angewandte Mathematik (in German), 68: 134–150, doi:10.1515/crll.1868.68.134, ISSN 0075-4102
• Szegő, Gábor (1939), Orthogonal Polynomials, Colloquium Publications, XXIII, American Mathematical Society, ISBN 978-0-8218-1023-1, MR 0372517