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 . The Laguerre polynomial can be defined by the contour integral
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 (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 (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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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.