Legendre polynomials > Catalogs

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

The first twelve Legendre polynomials are:

$\begin{align} P_0(x) &= 1 \\ P_1(x) &= x \\ P_2(x) &= \tfrac{1}{2}(3x^2-1)\\ P_3(x) &= \tfrac{1}{2}(5x^3 -3x)\\ P_4(x) &= \tfrac{1}{8}(35x^4- 30x^2 + 3)\\ P_5(x) &= \tfrac{1}{8}(63x^5- 70x^3 + 15x)\\ P_6(x) &= \tfrac{1}{16}(231x^6- 315x^4 + 105x^2 -5)\\ P_7(x) &= \tfrac{1}{16}(429x^7- 693x^5 + 315x^3 -35x)\\ P_8(x) &= \tfrac{1}{128}(6435x^8- 12012x^6 + 6930x^4 -1260x^2 + 35)\\ P_9(x) &= \tfrac{1}{128}(12155x^9- 25740x^7 + 18018x^5 -4620x^3 + 315x)\\ P_{10}(x) &= \tfrac{1}{256}(46189 x^{10}- 109395x^8 + 90090x^6 - 30030x^4 + 3465x^2 - 63 )\\ P_{11}(x) &= \tfrac{1}{256}( 88179x^{11}- 230945x^9 + 218790x^7 - 90090x^5 + 15015x^3 - 693x )\\ \end{align}$