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.

Legendre polynomials

The simplest case of the Mehler–Heine formula states that

\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 $J 0$ the Bessel function of order 0. The limit is uniform over $z$ in an arbitrary bounded domain in the complex plane.

Jacobi polynomials

The generalization to Jacobi polynomials $P (α, β) n$ is given by Gábor Szegő[3] as follows

\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 $α$ .

Laguerre polynomials

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

\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.

Hermite polynomials

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

{\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.

References

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

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[1] Mehler, G.F. (1868). "Ueber die Vertheilung der statischen Elektricität in einem von zwei Kugelkalotten begrenzten Körper" (PDF). Journal für die Reine und Angewandte Mathematik. 68: 134–150.

[2] Heine, E. (1861). Handbuch der Kugelfunktionen. Theorie und Anwendung. Berlin: Georg Reimer.

[3] Szegő, Gábor (1939). Orthogonal Polynomials. Colloquium Publications. American Mathematical Society. ISBN 978-0-8218-1023-1. MR 0372517.

[4] Koekoek, Roelof; Lesky, P.A.; Swarttouw, R.F. (2010). Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer-Verlag. doi:10.1007/978-3-642-05014-5. ISBN 978-3-642-05013-8.