Introduction E9F5FC

Questions FFFFC0

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Readings and links

• Wolfram Alpha: LegendreP[n,m,x] gives {$P_n^m(x)$}.
• Van Diejen, Kirillov. A Combinatorial Formula for the Associated Legendre Functions of Integer Degree.

Legendre polynomials

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

Rewriting the numerators and denominators

• {$P_{0}^{0}(x)=1$}
• {$P_{1}^{0}(x)=\frac{1}{2}(2x)$}
• {$P_{2}^{0}(x)=\frac{1}{4}(6x^{2}-2)$}
• {$P_{3}^{0}(x)=\frac{1}{8}(20x^3-12x)$}
• {$P_{4}^{0}(x)=\frac{1}{16}(70x^{4}-60x^{2}+6)$}
• {$P_{5}^{0}(x)=\frac{1}{32}(252x^{5}-280x^{3}+60x)$}
• {$P_{6}^{0}(x)=\frac{1}{64}(924x^{6}-1260x^{4}+420x^{2}-20)$}
• {$P_{7}^{0}(x)=\frac{1}{128}(3432x^{7}-5544x^{5}+2520x^{3}-280x)$}
• {$P_{8}^{0}(x)=\frac{1}{256}(12870x^{8}-24024x^{6}+13860x^{4}-2520x^{2}+70)$}
• {$P_{9}^{0}(x)={1\over 128}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x)$}
• {$P_{10}^{0}(x)={1\over 256}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)$}

The numerators of the Legendre polynomial are given by the ways of taking paths on a square grid from (0,0) to (n,n) where the possible steps are East (weight x), North (weight 1), and East-North-East (weight -1).

Associated Legendre Polynomials

{$$P_l^m(x)=(-1)^m(1-x^2)^{m/2}\frac{\textrm{d}^m}{\textrm{dx}^m}(P_l(x))$$}

P_0

{$P_{0}^{0}(x)=1$}

P_1

{$P_{1}^{-1}(x)=-\begin{matrix}\frac{1}{2}\end{matrix}P_{1}^{1}(x)$}

{$P_{1}^{0}(x)=x$}

{$P_{1}^{1}(x)=-(1-x^2)^{1/2}$}

P_2

{$P_{2}^{-2}(x)=\begin{matrix}\frac{1}{24}\end{matrix}P_{2}^{2}(x)$}

{$P_{2}^{-1}(x)=-\begin{matrix}\frac{1}{6}\end{matrix}P_{2}^{1}(x)$}

{$P_{2}^{0}(x)=\begin{matrix}\frac{1}{2}\end{matrix}(3x^{2}-1)$}

{$P_{2}^{1}(x)=-3x(1-x^2)^{1/2}$}

{$P_{2}^{2}(x)=3(1-x^2)$}

P_3

{$P_{3}^{-3}(x)=-\begin{matrix}\frac{1}{720}\end{matrix}P_{3}^{3}(x)$}

{$P_{3}^{-2}(x)=\begin{matrix}\frac{1}{120}\end{matrix}P_{3}^{2}(x)$}

{$P_{3}^{-1}(x)=-\begin{matrix}\frac{1}{12}\end{matrix}P_{3}^{1}(x)$}

{$P_{3}^{0}(x)=\begin{matrix}\frac{1}{2}\end{matrix}(5x^3-3x)$}

{$P_{3}^{1}(x)=-\begin{matrix}\frac{3}{2}\end{matrix}(5x^{2}-1)(1-x^2)^{1/2}$}

{$P_{3}^{2}(x)=15x(1-x^2)$}

{$P_{3}^{3}(x)=-15(1-x^2)^{3/2}$}

P_4

{$P_{4}^{-4}(x)=\begin{matrix}\frac{1}{40320}\end{matrix}P_{4}^{4}(x)$}

{$P_{4}^{-3}(x)=-\begin{matrix}\frac{1}{5040}\end{matrix}P_{4}^{3}(x)$}

{$P_{4}^{-2}(x)=\begin{matrix}\frac{1}{360}\end{matrix}P_{4}^{2}(x)$}

{$P_{4}^{-1}(x)=-\begin{matrix}\frac{1}{20}\end{matrix}P_{4}^{1}(x)$}

{$P_{4}^{0}(x)=\begin{matrix}\frac{1}{8}\end{matrix}(35x^{4}-30x^{2}+3)$}

{$P_{4}^{1}(x)=-\begin{matrix}\frac{5}{2}\end{matrix}(7x^3-3x)(1-x^2)^{1/2}$}

{$P_{4}^{2}(x)=\begin{matrix}\frac{15}{2}\end{matrix}(7x^2-1)(1-x^2)$}

{$P_{4}^{3}(x)= - 105x(1-x^2)^{3/2}$}

{$P_{4}^{4}(x)=105(1-x^2)^{2}$}

P_5

{$P_{5}^{-5}(x)={1\over 3840}\left(\sqrt{1-x^2}\right)^{5}$}

{$P_{5}^{-4}(x)={1\over 384}\left(\sqrt{1-x^2}\right)^{4}x$}

{$P_{5}^{-3}(x)={1\over 384}\left(\sqrt{1-x^2}\right)^{3}(9x^{2}-1)$}

{$P_{5}^{-2}(x)={1\over 16}\left(\sqrt{1-x^2}\right)^{2}(3x^{3}-1x)$}

{$P_{5}^{-1}(x)={1\over 16}\left(\sqrt{1-x^2}\right)(21x^{4}-14x^{2}+1)$}

{$P_{5}^{0}(x)={1\over 8}(63x^{5}-70x^{3}+15x)$}

{$P_{5}^{1}(x)={-15\over 8}\left(\sqrt{1-x^2}\right)(21x^{4}-14x^{2}+1)$}

{$P_{5}^{2}(x)={105\over 2}\left(\sqrt{1-x^2}\right)^{2}(3x^{3}-1x)$}

{$P_{5}^{3}(x)={-105\over 2}\left(\sqrt{1-x^2}\right)^{3}(9x^{2}-1)$}

{$P_{5}^{4}(x)=945\left(\sqrt{1-x^2}\right)^{4}x$}

{$P_{5}^{5}(x)=-945\left(\sqrt{1-x^2}\right)^{5}$}

P_6

{$P_{6}^{-6}(x)={1\over 46080}\left(\sqrt{1-x^2}\right)^{6}$}

{$P_{6}^{-5}(x)={1\over 3840}\left(\sqrt{1-x^2}\right)^{5}x$}

{$P_{6}^{-4}(x)={1\over 3840}\left(\sqrt{1-x^2}\right)^{4}(11x^{2}-1)$}

{$P_{6}^{-3}(x)={1\over 384}\left(\sqrt{1-x^2}\right)^{3}(11x^{3}-3x)$}

{$P_{6}^{-2}(x)={1\over 128}\left(\sqrt{1-x^2}\right)^{2}(33x^{4}-18x^{2}+1)$}

{$P_{6}^{-1}(x)={1\over 16}\left(\sqrt{1-x^2}\right)(33x^{5}-30x^{3}+5x)$}

{$P_{6}^{0}(x)={1\over 16}(231x^{6}-315x^{4}+105x^{2}-5)$}

{$P_{6}^{1}(x)={-21\over 8}\left(\sqrt{1-x^2}\right)(33x^{5}-30x^{3}+5x)$}

{$P_{6}^{2}(x)={105\over 8}\left(\sqrt{1-x^2}\right)^{2}(33x^{4}-18x^{2}+1)$}

{$P_{6}^{3}(x)={-315\over 2}\left(\sqrt{1-x^2}\right)^{3}(11x^{3}-3x)$}

{$P_{6}^{4}(x)={945\over 2}\left(\sqrt{1-x^2}\right)^{4}(11x^{2}-1)$}

{$P_{6}^{5}(x)=-10395\left(\sqrt{1-x^2}\right)^{5}x$}

{$P_{6}^{6}(x)=10395\left(\sqrt{1-x^2}\right)^{6}$}

P_7

{$P_{7}^{-7}(x)={1\over 645120}\left(\sqrt{1-x^2}\right)^{7}$}

{$P_{7}^{-6}(x)={1\over 46080}\left(\sqrt{1-x^2}\right)^{6}x$}

{$P_{7}^{-5}(x)={1\over 46080}\left(\sqrt{1-x^2}\right)^{5}(13x^{2}-1)$}

{$P_{7}^{-4}(x)={1\over 3840}\left(\sqrt{1-x^2}\right)^{4}(13x^{3}-3x)$}

{$P_{7}^{-3}(x)={1\over 3840}\left(\sqrt{1-x^2}\right)^{3}(143x^{4}-66x^{2}+3)$}

{$P_{7}^{-2}(x)={1\over 384}\left(\sqrt{1-x^2}\right)^{2}(143x^{5}-110x^{3}+15x)$}

{$P_{7}^{-1}(x)={1\over 128}\left(\sqrt{1-x^2}\right)(429x^{6}-495x^{4}+135x^{2}-5)$}

{$P_{7}^{0}(x)={1\over 16}(429x^{7}-693x^{5}+315x^{3}-35x)$}

{$P_{7}^{1}(x)={-7\over 16}\left(\sqrt{1-x^2}\right)(429x^{6}-495x^{4}+135x^{2}-5)$}

{$P_{7}^{2}(x)={63\over 8}\left(\sqrt{1-x^2}\right)^{2}(143x^{5}-110x^{3}+15x)$}

{$P_{7}^{3}(x)={-315\over 8}\left(\sqrt{1-x^2}\right)^{3}(143x^{4}-66x^{2}+3)$}

{$P_{7}^{4}(x)={3465\over 2}\left(\sqrt{1-x^2}\right)^{4}(13x^{3}-3x)$}

{$P_{7}^{5}(x)={-10395\over 2}\left(\sqrt{1-x^2}\right)^{5}(13x^{2}-1)$}

{$P_{7}^{6}(x)=135135\left(\sqrt{1-x^2}\right)^{6}x$}

{$P_{7}^{7}(x)=-135135\left(\sqrt{1-x^2}\right)^{7}$}

P_8

{$P_{8}^{-8}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{8}$}

{$P_{8}^{-7}(x)={1\over 645120}\left(\sqrt{1-x^2}\right)^{7}x$}

{$P_{8}^{-6}(x)={1\over 645120}\left(\sqrt{1-x^2}\right)^{6}(15x^{2}-1)$}

{$P_{8}^{-5}(x)={1\over 15360}\left(\sqrt{1-x^2}\right)^{5}(5x^{3}-1x)$}

{$P_{8}^{-4}(x)={1\over 15360}\left(\sqrt{1-x^2}\right)^{4}(65x^{4}-26x^{2}+1)$}

{$P_{8}^{-3}(x)={1\over 768}\left(\sqrt{1-x^2}\right)^{3}(39x^{5}-26x^{3}+3x)$}

{$P_{8}^{-2}(x)={1\over 256}\left(\sqrt{1-x^2}\right)^{2}(143x^{6}-143x^{4}+33x^{2}-1)$}

{$P_{8}^{-1}(x)={1\over 128}\left(\sqrt{1-x^2}\right)(715x^{7}-1001x^{5}+385x^{3}-35x)$}

{$P_{8}^{0}(x)={1\over 128}(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35)$}

{$P_{8}^{1}(x)={-9\over 16}\left(\sqrt{1-x^2}\right)(715x^{7}-1001x^{5}+385x^{3}-35x)$}

{$P_{8}^{2}(x)={315\over 16}\left(\sqrt{1-x^2}\right)^{2}(143x^{6}-143x^{4}+33x^{2}-1)$}

{$P_{8}^{3}(x)={-3465\over 8}\left(\sqrt{1-x^2}\right)^{3}(39x^{5}-26x^{3}+3x)$}

{$P_{8}^{4}(x)={10395\over 8}\left(\sqrt{1-x^2}\right)^{4}(65x^{4}-26x^{2}+1)$}

{$P_{8}^{5}(x)={-135135\over 2}\left(\sqrt{1-x^2}\right)^{5}(5x^{3}-1x)$}

{$P_{8}^{6}(x)={135135\over 2}\left(\sqrt{1-x^2}\right)^{6}(15x^{2}-1)$}

{$P_{8}^{7}(x)=-2027025\left(\sqrt{1-x^2}\right)^{7}x$}

{$P_{8}^{8}(x)=2027025\left(\sqrt{1-x^2}\right)^{8}$}

P_9

{$P_{9}^{-9}(x)={1\over 185794560}\left(\sqrt{1-x^2}\right)^{9}$}

{$P_{9}^{-8}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{8}x$}

{$P_{9}^{-7}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{7}(17x^{2}-1)$}

{$P_{9}^{-6}(x)={1\over 645120}\left(\sqrt{1-x^2}\right)^{6}(17x^{3}-3x)$}

{$P_{9}^{-5}(x)={1\over 215040}\left(\sqrt{1-x^2}\right)^{5}(85x^{4}-30x^{2}+1)$}

{$P_{9}^{-4}(x)={1\over 3072}\left(\sqrt{1-x^2}\right)^{4}(17x^{5}-10x^{3}+1x)$}

{$P_{9}^{-3}(x)={1\over 3072}\left(\sqrt{1-x^2}\right)^{3}(221x^{6}-195x^{4}+39x^{2}-1)$}

{$P_{9}^{-2}(x)={1\over 256}\left(\sqrt{1-x^2}\right)^{2}(221x^{7}-273x^{5}+91x^{3}-7x)$}

{$P_{9}^{-1}(x)={1\over 256}\left(\sqrt{1-x^2}\right)(2431x^{8}-4004x^{6}+2002x^{4}-308x^{2}+7)$}

{$P_{9}^{0}(x)={1\over 128}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x)$}

{$P_{9}^{1}(x)={-45\over 128}\left(\sqrt{1-x^2}\right)(2431x^{8}-4004x^{6}+2002x^{4}-308x^{2}+7)$}

{$P_{9}^{2}(x)={495\over 16}\left(\sqrt{1-x^2}\right)^{2}(221x^{7}-273x^{5}+91x^{3}-7x)$}

{$P_{9}^{3}(x)={-3465\over 16}\left(\sqrt{1-x^2}\right)^{3}(221x^{6}-195x^{4}+39x^{2}-1)$}

{$P_{9}^{4}(x)={135135\over 8}\left(\sqrt{1-x^2}\right)^{4}(17x^{5}-10x^{3}+1x)$}

{$P_{9}^{5}(x)={-135135\over 8}\left(\sqrt{1-x^2}\right)^{5}(85x^{4}-30x^{2}+1)$}

{$P_{9}^{6}(x)={675675\over 2}\left(\sqrt{1-x^2}\right)^{6}(17x^{3}-3x)$}

{$P_{9}^{7}(x)={-2027025\over 2}\left(\sqrt{1-x^2}\right)^{7}(17x^{2}-1)$}

{$P_{9}^{8}(x)=34459425\left(\sqrt{1-x^2}\right)^{8}x$}

{$P_{9}^{9}(x)=-34459425\left(\sqrt{1-x^2}\right)^{9}$}

P_10

{$P_{10}^{-10}(x)={1\over 3715891200}\left(\sqrt{1-x^2}\right)^{10}$}

{$P_{10}^{-9}(x)={1\over 185794560}\left(\sqrt{1-x^2}\right)^{9}x$}

{$P_{10}^{-8}(x)={1\over 185794560}\left(\sqrt{1-x^2}\right)^{8}(19x^{2}-1)$}

{$P_{10}^{-7}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{7}(19x^{3}-3x)$}

{$P_{10}^{-6}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{6}(323x^{4}-102x^{2}+3)$}

{$P_{10}^{-5}(x)={1\over 645120}\left(\sqrt{1-x^2}\right)^{5}(323x^{5}-170x^{3}+15x)$}

{$P_{10}^{-4}(x)={1\over 43008}\left(\sqrt{1-x^2}\right)^{4}(323x^{6}-255x^{4}+45x^{2}-1)$}

{$P_{10}^{-3}(x)={1\over 3072}\left(\sqrt{1-x^2}\right)^{3}(323x^{7}-357x^{5}+105x^{3}-7x)$}

{$P_{10}^{-2}(x)={1\over 3072}\left(\sqrt{1-x^2}\right)^{2}(4199x^{8}-6188x^{6}+2730x^{4}-364x^{2}+7)$}

{$P_{10}^{-1}(x)={1\over 256}\left(\sqrt{1-x^2}\right)(4199x^{9}-7956x^{7}+4914x^{5}-1092x^{3}+63x)$}

{$P_{10}^{0}(x)={1\over 256}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)$}

{$P_{10}^{1}(x)={-55\over 128}\left(\sqrt{1-x^2}\right)(4199x^{9}-7956x^{7}+4914x^{5}-1092x^{3}+63x)$}

{$P_{10}^{2}(x)={495\over 128}\left(\sqrt{1-x^2}\right)^{2}(4199x^{8}-6188x^{6}+2730x^{4}-364x^{2}+7)$}

{$P_{10}^{3}(x)={-6435\over 16}\left(\sqrt{1-x^2}\right)^{3}(323x^{7}-357x^{5}+105x^{3}-7x)$}

{$P_{10}^{4}(x)={45045\over 16}\left(\sqrt{1-x^2}\right)^{4}(323x^{6}-255x^{4}+45x^{2}-1)$}

{$P_{10}^{5}(x)={-135135\over 8}\left(\sqrt{1-x^2}\right)^{5}(323x^{5}-170x^{3}+15x)$}

{$P_{10}^{6}(x)={675675\over 8}\left(\sqrt{1-x^2}\right)^{6}(323x^{4}-102x^{2}+3)$}

{$P_{10}^{7}(x)={-11486475\over 2}\left(\sqrt{1-x^2}\right)^{7}(19x^{3}-3x)$}

{$P_{10}^{8}(x)={34459425\over 2}\left(\sqrt{1-x^2}\right)^{8}(19x^{2}-1)$}

{$P_{10}^{9}(x)=-654729075\left(\sqrt{1-x^2}\right)^{9}x$}

{$P_{10}^{10}(x)=654729075\left(\sqrt{1-x^2}\right)^{10}$}