Laguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by
- <math>
L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right).
<math>
These polynomials are orthogonal to each other with respect to the inner product given by
- <math>\langle f,g \rangle = \int_0^\infty f(x) g(x) e^{-x}\,dx.<math>
Generalization
The orthogonality property stated above is equivalent to saying that if X is an exponentially distributed random variable with probability density function
- <math>f(x)=\left\{\begin{matrix} f(x)=e^{-x} & \mbox{if}\ x>0, \\ 0 & \mbox{if}\ x<0, \end{matrix}\right\}<math>
then
- <math>E(L_n(X)L_m(X))=0\ \mbox{whenever}\ n\neq m.<math>
The exponential distribution is not the only gamma distribution. A polynomial sequence orthogonal with respect to the gamma distribution whose probabity density function is
- <math>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\}<math>
(see gamma function) is given by
- <math>L_n^{(\alpha)}(x)={x^{-\alpha} e^x \over n!}{d^n \over dx^n} e^{-x} x^{n+\alpha}.<math>
These are also sometimes called Laguerre polynomials. They coincide with the definition given above in case α = 0.
Sheffer sequence
The sequence of Laguerre polynomials is a Sheffer sequence.
