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

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

The first four Laguerre polynomials are:

$\left.L_{0}(x)\right.=1$

$\left.L_{1}(x)\right.=-x+1$

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_3 (x) =\frac{1}{6}(-x^3 +9x^2 -18x +6)}

Generalized Laguerre function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_n^{\alpha}(x)= \frac{(\alpha + 1)_n}{n!} ~_1F_1(-n; \alpha + 1;x)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a)_n} is the Pochhammer symbol and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~_1F_1(a;b;x)} is a confluent hyper-geometric function.

See also

• Laguerre Polynomial -- from Wolfram MathWorld