Mehler–Heine formula

Template:Short description In mathematics, the Mehler–Heine formula introduced by Gustav Ferdinand Mehler and Eduard Heine 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

$\underset{n\to \mathrm{\infty }}{\lim }{P}_n(\cos \frac{z}{n})=\underset{n\to \mathrm{\infty }}{\lim }{P}_n(1-\frac{z^2}{2n^2})=J_0(z),$

where Template:Math is the Legendre polynomial of order Template:Mvar, and Template:Math the Bessel function of order 0. The limit is uniform over Template:Mvar in an arbitrary bounded domain in the complex plane.

Jacobi polynomials

The generalization to Jacobi polynomials Template:Math is given by Gábor Szegő as follows

$\underset{n\to \mathrm{\infty }}{\lim }{n}^{-\alpha }{P}_n^{(\alpha ,\beta )}(\cos \frac{z}{n})=\underset{n\to \mathrm{\infty }}{\lim }{n}^{-\alpha }{P}_n^{(\alpha ,\beta )}(1-\frac{z^2}{2n^2})={\left(\frac{z}{2}\right)}^{-\alpha }{J}_{\alpha }(z),$

where Template:Math is the Bessel function of [[Bessel_function#Bessel_functions_of_the_first_kind:_Jα|order Template:Mvar]].

Laguerre polynomials

Using generalized Laguerre polynomials and confluent hypergeometric functions, they can be written as

$\underset{n\to \mathrm{\infty }}{\lim }{n}^{-\alpha }{L}_n^{(\alpha )}\left(\frac{z^2}{4n}\right)={\left(\frac{z}{2}\right)}^{-\alpha }{J}_{\alpha }(z),$

where Template:Math is the Laguerre function.

Hermite polynomials

Using the expressions equivalating Hermite polynomials and Laguerre polynomials where two equations exist, they can be written as

$\begin{array}{cc}\underset{n\to \mathrm{\infty }}{\lim }\frac{(-1{)}^n}{4^{n}n!}\sqrt{n}{H}_{2n}\left(\frac{z}{2\sqrt{n}}\right)& ={\left(\frac{z}{2}\right)}^{\frac{1}{2}}{J}_{-\frac{1}{2}}(z)\\ \underset{n\to \mathrm{\infty }}{\lim }\frac{(-1{)}^n}{4^{n}n!}{H}_{2n+1}\left(\frac{z}{2\sqrt{n}}\right)& =(2z{)}^{\frac{1}{2}}{J}_{\frac{1}{2}}(z),\end{array}$

where Template:Math is the Hermite function.