Laguerre polynomials

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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.

Also, for each n, Ln(x) is a solution of Laguerre's equation

(x D^2 + (1 - x) D + n) \, y(x) = 0

which is a second-order linear differential equation with variable coefficients.

The sequence of Laguerre polynomials is a Sheffer sequence.

Contents

Low orders

Image:Laguerre poly.svg
The first 5 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 xαe − x:

\int_0^{\infty}e^{-x}x^\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).

The associated Laguerre polynomials obey the following differential equation

xL_n^{(m) \prime\prime}(x) + (m+1-x)L_n^{(m)\prime}(x) + nL_n^{(m)}(x)=0.\,

Low-order examples of associated Laguerre polynomials

L_0^m(x) = 1
L_1^m(x) = -x + m +1
L_2^m(x) = \frac{x^2}{2} - (m + 2)x + \frac{(m+2)(m+1)}{2}

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! x L_n^{(1/2)} (x^2)

where the Hn(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 (which in this case represents the rising factorial).

References

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