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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 $xy'' + (1-x)y' + ny = 0$.

Comments

(5)

The $L_n$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{0}^{\infty} f(x)g(x) e^{-x} dx$.

Programs

(P1)

Sage

```
polynomials = {n: laguerre(n,x) for n in [0..100]}
```

Links

Data properties

Entries are of type: rational polynomial

Table is complete: no