Laguerre polynomials
From SklogWiki
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:
\[\left. L_0 (x) \right.=1\]
\[\left. L_1 (x) \right.=-x +1\]
\[L_2 (x) =\frac{1}{2}(x^2 -4x +2)\]
\[L_3 (x) =\frac{1}{6}(-x^3 +9x^2 -18x +6)\]
[edit] Generalized Laguerre function
\[L_n^{\alpha}(x)= \frac{(\alpha + 1)_n}{n!} ~_1F_1(-n; \alpha + 1;x)\]
where \((a)_n\) is the Pochhammer symbol and \( ~_1F_1(a;b;x)\) is a confluent hyper-geometric function.
