Laguerre polynomials

Laguerre polynomials are solutions $L_n(x)$ to the Laguerre differential equation with $\nu =0$. The Laguerre polynomial $H_n(z)$ can be defined by the contour integral

$L_n(z)=\frac{1}{2\pi i}\oint \frac{e^{-zt/(1-t)}}{(1-t)t^{n+1}}\mathrm{d}t$

The first four Laguerre polynomials are:

$L_0(x)=1$

$L_1(x)=-x+1$

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

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

Generalized Laguerre function

$L_n^{\alpha }(x)=\frac{(\alpha +1{)}_n}{n!}{}_1{F}_1(-n;\alpha +1;x)$

where $(a{)}_n$ is the Pochhammer symbol and ${}_1{F}_1(a;b;x)$ is a confluent hyper-geometric function.

See also

• Laguerre Polynomial -- from Wolfram MathWorld