Legendrovi polinomi

Legendrovi polinómi [ležándrovi ~] so rešitve Legendrove diferencialne enačbe:

${d \over dx}\left[(1-x^{2}){d \over dx}P_{n}(x)\right]+n(n+1)P_{n}(x)=0\!\,.$${d \over dx}\left[(1-x^{2}){d \over dx}P_{n}(x)\right]+n(n+1)P_{n}(x)=0\!\,.$

Imenovani so po Adrien-Marieu Legendru. Ta navadna diferencialna enačba je pogosto rabljena v fiziki in na drugih tehničnih področjih. Pojavi se pri reševanju Laplaceove enačbe in sorodnih parcialnih diferencialnih enačbah v sfernih koordinatah.

$P_{n}(x)={1 \over 2^{n}n!}{d^{n} \over dx^{n}}\left[(x^{2}-1)^{n}\right]\!\,.$$P_{n}(x)={1 \over 2^{n}n!}{d^{n} \over dx^{n}}\left[(x^{2}-1)^{n}\right]\!\,.$

Ortogonalnost

Pomembna lastnost Legendrovih polinomov je, da so ortogonalni v L² na intervalu −1 ≤ x ≤ 1:

$\int _{-1}^{1}P_{m}(x)P_{n}(x)\,dx={2 \over {2n+1}}\delta _{mn}$$\int _{-1}^{1}P_{m}(x)P_{n}(x)\,dx={2 \over {2n+1}}\delta _{mn}$,

(kjer je δ_mn oznaka za Kroneckerjev simbol delta, ki je 1, ko je m = n in 0 sicer).

Zgledi Legendrovih polinomov

Prvih nekaj Legendrovih polinomov:

n	$P_{n}(x)\,$$P_{n}(x)\,$
0	$1\,$$1\,$
1	$x\,$$x\,$
2	${\begin{matrix}{\frac {1}{2}}\end{matrix}}(3x^{2}-1)\,$${\begin{matrix}{\frac {1}{2}}\end{matrix}}(3x^{2}-1)\,$
3	${\begin{matrix}{\frac {1}{2}}\end{matrix}}(5x^{3}-3x)\,$${\begin{matrix}{\frac {1}{2}}\end{matrix}}(5x^{3}-3x)\,$
4	${\begin{matrix}{\frac {1}{8}}\end{matrix}}(35x^{4}-30x^{2}+3)\,$${\begin{matrix}{\frac {1}{8}}\end{matrix}}(35x^{4}-30x^{2}+3)\,$
5	${\begin{matrix}{\frac {1}{8}}\end{matrix}}(63x^{5}-70x^{3}+15x)\,$${\begin{matrix}{\frac {1}{8}}\end{matrix}}(63x^{5}-70x^{3}+15x)\,$
6	${\begin{matrix}{\frac {1}{16}}\end{matrix}}(231x^{6}-315x^{4}+105x^{2}-5)\,$${\begin{matrix}{\frac {1}{16}}\end{matrix}}(231x^{6}-315x^{4}+105x^{2}-5)\,$
7	${\begin{matrix}{\frac {1}{16}}\end{matrix}}(429x^{7}-693x^{5}+315x^{3}-35x)\,$${\begin{matrix}{\frac {1}{16}}\end{matrix}}(429x^{7}-693x^{5}+315x^{3}-35x)\,$
8	${\begin{matrix}{\frac {1}{128}}\end{matrix}}(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35)\,$${\begin{matrix}{\frac {1}{128}}\end{matrix}}(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35)\,$
9	${\begin{matrix}{\frac {1}{128}}\end{matrix}}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x)\,$${\begin{matrix}{\frac {1}{128}}\end{matrix}}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x)\,$
10	${\begin{matrix}{\frac {1}{256}}\end{matrix}}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)\,$${\begin{matrix}{\frac {1}{256}}\end{matrix}}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)\,$

Glej tudi

• polinomi Čebišova