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  "body": "\n## Definition\n\nThe *Legendre polynomials* are the real polynomials $P_n$, for $n \\ge 0$, determined by the following conditions:\n\n1. $P_n$ has [[degree|0P4JRMSWNH]] exactly $n$.\n2. $\\int_{-1}^{1} P_m(x)\\, P_n(x)\\, dx = 0$ for all $m \\ne n$.\n3. $P_n(1) = 1$.\n\n## Remarks\n\nConditions 1 and 2 fix $P_n$ up to a constant factor, and condition 3 fixes the factor.\nExample: $P_2(x) = \\frac{3x^2 - 1}{2}$; it has degree $2$, $P_2(1) = 1$, and $\\int_{-1}^1 P_2\\, dx = \\frac{1}{2}\\left( 2 - 2 \\right) = 0$ and $\\int_{-1}^1 x P_2\\, dx = 0$ since $x P_2$ is odd.\nA non-example: $x^2$ has degree $2$ and value $1$ at $1$, but $\\int_{-1}^1 x^2 \\cdot 1\\, dx = \\frac{2}{3} \\ne 0$, so it fails condition 2 against $P_0 = 1$.\nThey are given by $P_n(x) = \\frac{1}{2^n n!} \\frac{d^n}{dx^n} (x^2 - 1)^n$, and they satisfy $(n + 1) P_{n+1}(x) = (2n + 1)\\, x P_n(x) - n P_{n-1}(x)$ and $\\int_{-1}^1 P_n(x)^2\\, dx = \\frac{2}{2n + 1}$.\nTheir generating function is $\\frac{1}{\\sqrt{1 - 2xt + t^2}} = \\sum_{n \\ge 0} P_n(x)\\, t^n$ for $\\abs{t}$ small.\nThey are the [[classical orthogonal polynomials|47WJ34NC9K]] of Jacobi type with $\\alpha = \\beta = 0$, and $P_n$ solves $(1 - x^2) y'' - 2x y' + n(n+1) y = 0$.\n",
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