Laguerre polynomials

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Laguerre polynomials are solutions Ln(x) to the Laguerre differential equation with ν = 0. The Laguerre polynomial Hn(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.

[edit] See also

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