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.

The simplest case of the Mehler–Heine formula states that

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.

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

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