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In mathematics, the '''Mehler–Heine formula''' introduced by {{harvs|txt|authorlink=G. F. Mehler|last=Mehler|year=1868}} and {{harvs|txt|authorlink=Eduard  Heine|last=Heine|year=1861}} 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
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:<math>\lim _{n\to\infty}P_n\Bigl(\cos{z\over n}\Bigr)=J_0(z)</math>
 
where ''P''<sub>''n''</sub> is the Legendre polynomial of order ''n'', and ''J''<sub>0</sub> a [[Bessel function]]. The limit is uniform over ''z'' in an arbitrary bounded [[domain (mathematical analysis)|domain]] in the [[complex plane]].
 
==Jacobi polynomials==
 
The generalization to [[Jacobi polynomials]] ''P''{{su|b=''n''|p=α,β}} is given by {{harv|Szegő|1939|loc=8.1}} as follows:
 
:<math>\lim_{n \to \infty} n^{-\alpha}P_n^{\alpha,\beta}\left(\cos \frac{z}{n}\right)
= \left(\frac{z}{2}\right)^{-\alpha} J_\alpha(z)~. </math>
 
==References==
 
*{{Citation | last1=Heine | first1=E. | title=Handbuch der Kugelfunktionen. Theorie und Anwendung. Neudruck. | url=http://books.google.com/books?id=D79hEMl2GM0C | publisher=Georg Reimer, Berlin | language=German | zbl=0103.29304 | year=1861 }}
*{{Citation | last1=Mehler | first1=F. G. | title=Ueber die Vertheilung der statischen Elektricität in einem von zwei Kugelkalotten begrenzten Körper. | language=German | doi=10.1515/crll.1868.68.134 | year=1868 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=68 | pages=134–150}}
*{{Citation | last1=Szegő | first1=Gábor | title=Orthogonal Polynomials | url=http://books.google.com/books?id=3hcW8HBh7gsC | publisher= American Mathematical Society | series=Colloquium Publications | isbn=978-0-8218-1023-1 | mr=0372517 | year=1939 | volume=XXIII}}
 
{{DEFAULTSORT:Mehler-Heine formula}}
[[Category:Orthogonal polynomials]]

Latest revision as of 05:12, 26 May 2014


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