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Legendre polynomials
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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