Laguerre L

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The Laguerre polynomial of order $n$, $L_n$, is given by $$L_n(x) = \displaystyle\sum_{k=0}^n \dfrac{(-1)^kn!}{(n-k)!(k!)^2}x^k.$$

The first few Laguerre polynomials are given by $$\begin{array}{ll} L_0(x) &= 1 \\ L_1(x) &= -x+1 \\ L_2(x) &= \dfrac{1}{2}(x^2-4x+2) \\ L_3(x) &= \dfrac{1}{6}(-x^3+9x^2-18x+6) \\ L_4(x) &= \dfrac{1}{24}(x^4-16x^3+72x^2-96x+24)\\ \vdots \end{array}$$


Generating function for Laguerre L
L n(x)=(e^x/n!)d^n/dx^n(x^n e^(-x))
L n(0)=1
L n'(0)=-n
Orthogonality of Laguerre L
(n+1)L (n+1)(x) = (2n+1-x)L n(x)-nL (n-1)(x)
xL n'(x)=nL n(x)-n L (n-1)(x)
L n'(x)=-Sum L k(x)

See also

Associated Laguerre L


Orthogonal polynomials