Laguerre polynomials

Laguerre polynomials are solutions $L_{n}(x)$ to the Laguerre differential equation with $\nu =0$. The Laguerre polynomial Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.L_{0}(x)\right.=1}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.L_{1}(x)\right.=-x+1}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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

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 $(a)_{n}$ is the Pochhammer symbol and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ~_{1}F_{1}(a;b;x)} is a confluent hyper-geometric function.

See also

• Laguerre Polynomial -- from Wolfram MathWorld