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 External links

7 References

Low orders

Missing image
Laguerre-polynomials.PNG

The first 6 Laguerre polynomials.

The first few polynomials are

$L_0(x)=1\,$

$L_1(x)=-x+1\,$

$L_2(x)=\frac{1}{2}(x^2-4x+2)$

$L_3(x)=\frac{1}{6}(-x^3+9x^2-18x+6)$

$L_4(x)=\frac{1}{24}(x^4-16x^3+72x^2-96x+24)$

$L_5(x)=\frac{1}{120}(-x^5+25x^4-200x^3+600x^2-600x+120)$

$L_6(x)=\frac{1}{720}(x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720)$

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 (rising factorial).

External links

Laguerre polynomial on MathWorld

References

Milton Abramowitz and Irene A. Stegun, eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0486612724. (See chapter 22)