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Laguerre polynomials
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Polynomials
$n$
$L_n(x)$
INPUT{polynomials.yaml} (not shown in preview)
Definition
The Laguerre polynomials $L_n$, $n\geq 0$, can be defined as the system of orthogonal polynomials with respect to the inner product (5) such that $\deg L_n=n$ and $L_n(0)=1$.
Parameters
$n$
—   integer ($n \geq 0$)
Formulas
(1)
$L_0(x) = 1$, $L_1(x) = 1-x$, and $(n+1)L_{n+1}(x) = (2n+1-x) L_n(x) - n L_{n-1}(x)$ for $n\geq 1$ (recurrence formula).
(2)
$L_n(x) = \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{k!} x^k$ (closed form).
(3)
$\sum_{n=0}^\infty L_n(x)t^n = \frac{1}{1-t} e^{-tx/(1-t)}$ (generating function)..
(4)
$L_n$ is a solution of Laguerre's equation $xy'' + (1-x)y' + ny = 0$.
The $L_n$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{0}^{\infty} f(x)g(x) e^{-x} dx$.
polynomials = {n: laguerre(n,x) for n in [0..100]}