# Legendre polynomials/Catalogs

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An informational catalog, or several catalogs, about Legendre polynomials.

The first twelve Legendre polynomials are:

{\displaystyle {\begin{aligned}P_{0}(x)&=1\\P_{1}(x)&=x\\P_{2}(x)&={\tfrac {1}{2}}(3x^{2}-1)\\P_{3}(x)&={\tfrac {1}{2}}(5x^{3}-3x)\\P_{4}(x)&={\tfrac {1}{8}}(35x^{4}-30x^{2}+3)\\P_{5}(x)&={\tfrac {1}{8}}(63x^{5}-70x^{3}+15x)\\P_{6}(x)&={\tfrac {1}{16}}(231x^{6}-315x^{4}+105x^{2}-5)\\P_{7}(x)&={\tfrac {1}{16}}(429x^{7}-693x^{5}+315x^{3}-35x)\\P_{8}(x)&={\tfrac {1}{128}}(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35)\\P_{9}(x)&={\tfrac {1}{128}}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x)\\P_{10}(x)&={\tfrac {1}{256}}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)\\P_{11}(x)&={\tfrac {1}{256}}(88179x^{11}-230945x^{9}+218790x^{7}-90090x^{5}+15015x^{3}-693x)\\\end{aligned}}}