Mehler–Heine formula

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 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

${\displaystyle \lim _{n\to \infty }P_{n}\left(\cos {\frac {z}{n}}\right)=\lim _{n\to \infty }P_{n}\left(1-{\frac {z^{2}}{2n^{2}}}\right)=J_{0}(z),}$

where P_n is the Legendre polynomial of order n, and J₀ the Bessel function of order 0. 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

${\displaystyle \lim _{n\to \infty }n^{-\alpha }P_{n}^{(\alpha ,\beta )}\left(\cos {\frac {z}{n}}\right)=\lim _{n\to \infty }n^{-\alpha }P_{n}^{(\alpha ,\beta )}\left(1-{\frac {z^{2}}{2n^{2}}}\right)=\left({\frac {z}{2}}\right)^{-\alpha }J_{\alpha }(z),}$

where J_α is the Bessel function of order α.

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

${\displaystyle \lim _{n\to \infty }n^{-\alpha }L_{n}^{(\alpha )}\left({\frac {z^{2}}{4n}}\right)=\left({\frac {z}{2}}\right)^{-\alpha }J_{\alpha }(z),}$

where L^(α)
_n is the Laguerre function.

Using the expressions equivalating Hermite polynomials and Laguerre polynomials where two equations exist,^[4] they can be written as

${\displaystyle {\begin{aligned}\lim _{n\to \infty }{\frac {(-1)^{n}}{4^{n}n!}}{\sqrt {n}}H_{2n}\left({\frac {z}{2{\sqrt {n}}}}\right)&=\left({\frac {z}{2}}\right)^{\frac {1}{2}}J_{-{\frac {1}{2}}}(z)\\\lim _{n\to \infty }{\frac {(-1)^{n}}{4^{n}n!}}H_{2n+1}\left({\frac {z}{2{\sqrt {n}}}}\right)&=(2z)^{\frac {1}{2}}J_{\frac {1}{2}}(z),\end{aligned}}}$

where H_n is the Hermite function.