Laguerre polynomials

Laguerre polynomials are solutions Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{n}(x)} to the Laguerre differential equation with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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:

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 \left. L_0 (x) \right.=1}

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 \left. L_1 (x) \right.=-x +1}

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_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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{n}^{\alpha }(x)={\frac {(\alpha +1)_{n}}{n!}}~_{1}F_{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