In mathematics, the Mehler–Heine formula introduced by Gustav Ferdinand Mehler[1] and Eduard Heine[2] 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 limit is uniform over z in an arbitrary bounded domain in the complex plane.
The generalization to Jacobi polynomials P(α, β)n is given by Gábor Szegő[3] as follows where Jα is the Bessel function of order α.
Using the expressions equivalating Hermite polynomials and Laguerre polynomials where two equations exist,[4] they can be written as where Hn is the Hermite function.