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  "body": "\n## Definition\n\nLet $\\alpha, \\beta \u003e -1$ be real numbers and let $n \\ge 0$ be an integer.\nWrite $(a)_k = a (a + 1) \\cdots (a + k - 1)$ for the [[Pochhammer symbol|YZHM3HSM0F]], with $(a)_0 = 1$.\nThe *Jacobi polynomial* of degree $n$ with parameters $\\alpha, \\beta$ is\n$$P_n^{(\\alpha, \\beta)}(x) = \\sum_{k=0}^{n} \\frac{(n + \\alpha + \\beta + 1)_k \\, (\\alpha + k + 1)_{n-k}}{k! \\, (n - k)!} \\left( \\frac{x - 1}{2} \\right)^k.$$\n\n## Remarks\n\nEquivalently, $P_n^{(\\alpha, \\beta)}(x) = \\frac{(\\alpha + 1)_n}{n!} \\, {}_2F_1\\left( -n, n + \\alpha + \\beta + 1; \\alpha + 1; \\frac{1 - x}{2} \\right)$.\nRodrigues' formula: $(1 - x)^\\alpha (1 + x)^\\beta P_n^{(\\alpha, \\beta)}(x) = \\frac{(-1)^n}{2^n n!} \\frac{d^n}{dx^n} \\left( (1 - x)^{\\alpha + n} (1 + x)^{\\beta + n} \\right)$.\nThey are orthogonal: $\\int_{-1}^{1} P_m^{(\\alpha, \\beta)}(x) P_n^{(\\alpha, \\beta)}(x) (1 - x)^\\alpha (1 + x)^\\beta \\, dx = 0$ for $m \\ne n$.\nExample: $P_1^{(\\alpha, \\beta)}(x) = (\\alpha + 1) + (\\alpha + \\beta + 2) \\frac{x - 1}{2}$; for $\\alpha = \\beta = 0$ this is $x$, the Legendre polynomial $P_1$.\nThe value at $1$ is $P_n^{(\\alpha, \\beta)}(1) = \\frac{(\\alpha + 1)_n}{n!}$, since only the term $k = 0$ survives.\nNon-example: for $\\alpha = -1$ the weight $(1 - x)^{-1}$ is not integrable near $x = 1$, so the clause $\\alpha \u003e -1$ fails and the orthogonality integrals diverge.\n",
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