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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.

[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.

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+{{Short description|Formula describing the asymptotic behavior of the Legendre polynomials}}
-In mathematics, the '''Mehler–Heine formula''' describes the asymptotic behavior of the [[orthogonal polynomials#The classical orthogonal polynomials|classical orthogonal polynomials]] as the index tends to infinity, near the edges of the support of the weight. The formula complements the Darboux formulae which describes the asymptotics in the interior and outside the support.
+In mathematics, the '''Mehler–Heine formula''' introduced by [[Gustav Ferdinand Mehler]]<ref>{{Cite journal |last=Mehler |first=G.F. |date=1868 |title=Ueber die Vertheilung der statischen Elektricität in einem von zwei Kugelkalotten begrenzten Körper |url=https://zenodo.org/record/1448892/files/article.pdf |journal=Journal für die Reine und Angewandte Mathematik |volume=68 |pages=134-150}}</ref> and [[Eduard Heine]]<ref>{{Cite book |last=Heine |first=E. |url=https://books.google.com/books?id=D79hEMl2GM0C |title=Handbuch der Kugelfunktionen. Theorie und Anwendung. |publisher=Georg Reimer |year=1861 |location=Berlin}}</ref> 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.
-For [[Legendre polynomials]] it was introduced by [[G. F. Mehler]] and [[Eduard  Heine]].
 ==Legendre polynomials==
 The simplest case of the Mehler–Heine formula states that
-:<math>\lim _{n\to\infty}P_n\Bigl(\cos{\theta\over n}\Bigr)=J_0(\theta)</math>
+:<math>\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),</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 ''&theta;'' in an arbitrary bounded [[domain (mathematical analysis)|domain]] in the [[complex plane]].
+where {{math|''P''<sub>''n''</sub>}} is the Legendre polynomial of order {{mvar|n}}, and {{math|''J''<sub>0</sub>}} the [[Bessel function]] of order 0. The limit is uniform over {{mvar|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:
+The generalization to [[Jacobi polynomials]] {{math|''P''{{su|b=''n''|p=(''&alpha;'', ''&beta;'')}}}} is given by [[Gábor Szegő]]<ref>{{Cite book |last=Szegő |first=Gábor |title=Orthogonal Polynomials |publisher=American Mathematical Society |year=1939 |isbn=978-0-8218-1023-1 |series=Colloquium Publications |mr=0372517}}</ref> as follows
-:<math>\lim_{n \to \infty} n^{-\alpha}P_n^{\alpha,\beta}\left(\cos \frac{z}{n}\right)
+:<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>
+= \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), </math>
+where {{math|''J''<sub>''&alpha;''</sub>}} is the Bessel function of [[Bessel_function#Bessel_functions_of_the_first_kind:_Jα|order {{mvar|''&alpha;''}}]].
-==References==
+==Laguerre polynomials==
-*{{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 | id={{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}}
+Using generalized [[Laguerre polynomials]] and [[confluent hypergeometric function|confluent hypergeometric functions]], they can be written as
-*{{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 | id={{MathSciNet | id = 0372517}} | year=1939 | volume=XXIII}}
+:<math>\lim_{n \to \infty} n^{-\alpha}L_n^{(\alpha)}\left(\frac{z^2}{4n}\right)
+= \left(\frac{z}{2}\right)^{-\alpha} J_\alpha(z),</math>
+where {{math|''L''{{su|b=''n''|p=(''&alpha;'')}}}} is the Laguerre function.
+==Hermite polynomials==
+Using the expressions equivalating [[Hermite polynomials]] and Laguerre polynomials where [[Hermite polynomials#Laguerre polynomials|two equations]] exist,<ref>{{Cite book |last=Koekoek |first=Roelof |title=Hypergeometric Orthogonal Polynomials and Their q-Analogues |last2=Lesky |first2=P.A. |last3=Swarttouw |first3=R.F. |publisher=Springer-Verlag |year=2010 |isbn=978-3-642-05013-8 |doi=10.1007/978-3-642-05014-5}}</ref> they can be written as
+:<math>\begin{align}\lim_{n \to \infty} \frac{(-1)^n}{4^nn!}\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^nn!}H_{2n+1}\left(\frac{z}{2\sqrt{n}}\right)
+&=(2z)^{\frac{1}{2}}J_{\frac{1}{2}}(z),\end{align}</math>
+where {{math|''H''<sub>''n''</sub>}} is the Hermite function.
+==References==
+{{Reflist}}
+{{DEFAULTSORT:Mehler-Heine formula}}
 [[Category:Orthogonal polynomials]]