Mehler–Heine formula explained

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.

Legendre polynomials

The simplest case of the Mehler–Heine formula states that

\lim _{n\toinfty} P _{n\left(\cos{ (z)/(n)} }\right)= \lim _P_n\left(1-\frac\right)= J_0(z),

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

Jacobi polynomials

The generalization to Jacobi polynomials is given by Gábor Szegő^[3] as follows

\lim _{n} n ^{-\alpha} P ^{(\alpha,\beta)} _{n} \left(\cos (z)/(n) \right) = \lim _{n} n ^{-\alpha} P ^{(\alpha,\beta)} _{n} \left( 1- (z ^{2})/(2 n ^{2}) \right) = \left( (z)/(2) \right) ^{-\alpha} J _{\alpha(z),}

where is the Bessel function of order .

Laguerre polynomials

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

\lim _{n} n ^{-\alpha} L ^{(\alpha)} _{n} \left( (z ^{2})/(4 n) \right) = \left( (z)/(2) \right) ^{-\alpha} J _{\alpha(z),}

where is the Laguerre function.

Hermite polynomials

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

\begin{align}\lim _{n} (( -1 ) ^{n})/(4 ^{nn!}) \sqrt{n}H _{2n} \left( (z)/(2 \sqrt{n) }\right)&=\left(\frac\right)^J_(z) \\\lim_ \fracH_\left(\frac\right)&=(2z)^J_(z),\end

where is the Hermite function.

Notes and References

1. Mehler . G.F. . 1868 . Ueber die Vertheilung der statischen Elektricität in einem von zwei Kugelkalotten begrenzten Körper . Journal für die Reine und Angewandte Mathematik . 68 . 134-150.
2. Book: Heine, E. . Handbuch der Kugelfunktionen. Theorie und Anwendung. . Georg Reimer . 1861 . Berlin.
3. Book: Szegő, Gábor . Orthogonal Polynomials . American Mathematical Society . 1939 . 978-0-8218-1023-1 . Colloquium Publications . 0372517.
4. Book: Koekoek, Roelof . Hypergeometric Orthogonal Polynomials and Their q-Analogues . Lesky . P.A. . Swarttouw . R.F. . Springer-Verlag . 2010 . 978-3-642-05013-8 . 10.1007/978-3-642-05014-5.