Mehler–Heine formula

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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[edit]

The simplest case of the Mehler–Heine formula states that

\lim _{n\to\infty}P_n\Bigl(\cos{z\over n}\Bigr)=J_0(z)

where Pn 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[edit]

The generalization to Jacobi polynomials Pα,β
is given by (Szegő 1939, 8.1) as follows:

\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)~.