— preview —

Legendre polynomials

edit on github
Polynomials

$n$

$P_n(x)$

INPUT{polynomials.yaml} (not shown in preview)

Definition

The Legendre polynomials $P_n$, $n\geq 0$, can be defined as the system of orthogonal polynomials with respect to the inner product (3) such that $\deg P_n=n$ and $P_n(1)=1$.

Parameters

$n$

— integer ($n \geq 0$)

Formulas

(1)

$P_0(x) = 1$, $P_1(x) = x$, and $(n+1)P_{n+1}(x) = (2n+1)_x P_n(x) - n P_{n-1}(x)$ for $n\geq 1$ (Bonnet's recursion formula).

(2)

$\sum_{n=0}^\infty P_n(x)t^n = \frac{1}{\sqrt{1-2tx+t^2}}$ (generating function).

Comments

(3)

The $P_n$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) dx$.

Programs

(P1)

Sage

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

Links

Data properties

Entries are of type: rational polynomial

Table is complete: no