چندجملهای های لژاندر (Legendre polynomials) جوابهایِ معادله دیفرانسیل معمولی زیر موسوم به معادلهٔ دیفرانسیل لژاندر هستند:
d d x [ ( 1 − x 2 ) d d x P n ( x ) ] + n ( n + 1 ) P n ( x ) = 0. {\displaystyle {d \over dx}\left[(1-x^{2}){d \over dx}P_{n}(x)\right]+n(n+1)P_{n}(x)=0.} مثالها اولین چندجملهایهای لژاندر به صورت زیر هستند:
n P n ( x ) {\displaystyle P_{n}(x)\,} 0 1 {\displaystyle 1\,} 1 x {\displaystyle x\,} 2 1 2 ( 3 x 2 − 1 ) {\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}(3x^{2}-1)\,} 3 1 2 ( 5 x 3 − 3 x ) {\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}(5x^{3}-3x)\,} 4 1 8 ( 35 x 4 − 30 x 2 + 3 ) {\displaystyle {\begin{matrix}{\frac {1}{8}}\end{matrix}}(35x^{4}-30x^{2}+3)\,} 5 1 8 ( 63 x 5 − 70 x 3 + 15 x ) {\displaystyle {\begin{matrix}{\frac {1}{8}}\end{matrix}}(63x^{5}-70x^{3}+15x)\,} 6 1 16 ( 231 x 6 − 315 x 4 + 105 x 2 − 5 ) {\displaystyle {\begin{matrix}{\frac {1}{16}}\end{matrix}}(231x^{6}-315x^{4}+105x^{2}-5)\,} 7 1 16 ( 429 x 7 − 693 x 5 + 315 x 3 − 35 x ) {\displaystyle {\begin{matrix}{\frac {1}{16}}\end{matrix}}(429x^{7}-693x^{5}+315x^{3}-35x)\,} 8 1 128 ( 6435 x 8 − 12012 x 6 + 6930 x 4 − 1260 x 2 + 35 ) {\displaystyle {\begin{matrix}{\frac {1}{128}}\end{matrix}}(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35)\,} 9 1 128 ( 12155 x 9 − 25740 x 7 + 18018 x 5 − 4620 x 3 + 315 x ) {\displaystyle {\begin{matrix}{\frac {1}{128}}\end{matrix}}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x)\,} 10 1 256 ( 46189 x 10 − 109395 x 8 + 90090 x 6 − 30030 x 4 + 3465 x 2 − 63 ) {\displaystyle {\begin{matrix}{\frac {1}{256}}\end{matrix}}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)\,}
پیوند به بیرون منابع Abramowitz, Milton & Stegun, Irene A., eds. (1965), “Chapter 8”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 See also chapter 22.
![{\displaystyle {d \over dx}\left[(1-x^{2}){d \over dx}P_{n}(x)\right]+n(n+1)P_{n}(x)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd51c7228067db4bea119843fb19c6caab834954)
