In mathematics, the Mehler–Heine formula introduced by

Gustav Ferdinand Mehler Gustav Ferdinand Mehler, or Ferdinand Gustav Mehler (13 December 1835, in Schönlanke, Kingdom of Prussia – 13 July 1895, in Elbing, German Empire) was a German mathematician. He is credited with introducing Mehler's formula; the Mehler–Foc ...

and Eduard Heine describes the asymptotic behavior of the

Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...

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 mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as bec ...

in the interior and outside the support.

Legendre polynomials

The simplest case of the Mehler–Heine formula states that :

\lim _P_n\left(\cos\right)
= \lim _P_n\left(1-\frac\right)
= J_0(z),

where is the Legendre polynomial of order , and the

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

of order 0. The limit is uniform over in an arbitrary bounded

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

Jacobi polynomials

The generalization to Jacobi polynomials is given by

Gábor Szegő Gábor Szegő () (January 20, 1895 – August 7, 1985) was a Hungarian-American mathematician. He was one of the foremost mathematical analysts of his generation and made fundamental contributions to the theory of orthogonal polynomials and T ...

as follows :

\lim_ n^P_n^\left(\cos \frac\right)
= \lim_ n^P_n^\left(1-\frac\right)
= \left(\frac\right)^ J_\alpha(z),

where is the Bessel function of

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

Laguerre polynomials

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

\lim_ n^L_n^\left(\frac\right)
= \left(\frac\right)^ J_\alpha(z),

where is the Laguerre function.

Hermite polynomials

Using the expressions equivalating

Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...

and Laguerre polynomials where two equations exist, they can be written as :

\begin\lim_ \frac\sqrtH_\left(\frac\right)
&=\left(\frac\right)^J_(z) \\
\lim_ \fracH_\left(\frac\right)
&=(2z)^J_(z),\end

where is the Hermite function.

References

{{DEFAULTSORT:Mehler-Heine formula Orthogonal polynomials