Chemistry - Laguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by

$L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right).$

These polynomials are orthogonal to each other with respect to the inner product given by

$\langle f,g \rangle = \int_0^\infty f(x) g(x) e^{-x}\,dx.$

The sequence of Laguerre polynomials is a Sheffer sequence.

Contents

1 Low orders

2 As contour integral

3 Generalized Laguerre polynomials

4 Relation to Hermite polynomials

5 Relation to hypergeometric functions

6 References

Low orders

The first few polynomials are

L 0 (x) = 1

L 1 (x) = - x + 1

$L_2(x)=\frac{1}{2}x^2-2x+1.$

As contour integral

The polynomials may be expressed in terms of a contour integral

$L_n(x)=\frac{1}{2\pi i}\oint\frac{e^{-(xt)/(1-t)}}{(1-t)\,t^{n+1}} \; dt$

where the contour circles the origin once in a counterclockwise direction.

Generalized Laguerre polynomials

The orthogonality property stated above is equivalent to saying that if X is an exponentially distributed random variable with probability density function

$f(x)=\left\{\begin{matrix} f(x)=e^{-x} & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right\}$

then

$E(L_n(X)L_m(X))=0\ \mbox{whenever}\ n\neq m.$

The exponential distribution is not the only gamma distribution. A polynomial sequence orthogonal with respect to the gamma distribution whose probability density function is

$f(x)=\left\{\begin{matrix} f(x)=x^{\alpha-1} e^{-x}/\Gamma(\alpha) & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right\}$

(see gamma function) is given by the defining equation for the generalized Laguerre polynomials:

$L_n^{(\alpha)}(x)= {x^{-\alpha} e^x \over n!}{d^n \over dx^n} e^{-x} x^{n+\alpha}.$

These are also sometimes called the associated Laguerre polynomials. The simple Laguerre polynomials are recovered from the generalized polynomials by setting α=0:

$L^{(0)}_n(x)=L_n(x).$

The associated Laguerre polynomials are orthogonal over $[0,\infty)$ with respect to the weighting function $x α e - x$ :

$\int_0^{\infty}e^xx^\alpha L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)dx=\frac{\Gamma(n+\alpha+1)}{n!}\delta_{nm}.$

For integer α the defining equation above can be written as

$L_n^{(m)}(x)= (-1)^m{d^m \over dx^m} L_{n+m}(x).$

Relation to Hermite polynomials

The generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator, due to their relation to the Hermite polynomials, which can be expressed as

$H_{2n}(x) = (-1)^n 2^{2n} n! L_n^{(-1/2)} (x^2)$

and

$H_{2n+1}(x) = (-1)^n 2^{2n+1} n! L_n^{(1/2)} (x^2)$

where the $H n (x)$ are the Hermite polynomials.

Relation to hypergeometric functions

The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as

$L^a_n(x) = {n+a \choose n} M(-n,a+1,x) =\frac{(a+1)_n} {n!} \,_1F_1(-n,a+1,x)$

where $(a) n$ is the Pochhammer symbol.

References

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . (See chapter 22).
Laguerre polynomial on MathWorld

Categories: Polynomials | Special functions