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  "body": "\n## Definition\n\nLet $\\lambda \u003e -\\frac12$ be a real number with $\\lambda \\ne 0$.\nThe *Gegenbauer polynomials* $C_n^{(\\lambda)}$, $n \\ge 0$, are the polynomials defined by the [[generating function|0PHAC833JF]]\n$$\\frac{1}{(1 - 2xt + t^2)^{\\lambda}} = \\sum_{n=0}^{\\infty} C_n^{(\\lambda)}(x) \\, t^n,$$\nvalid for $x \\in [-1, 1]$ and $\\abs{t} \u003c 1$.\n\n## Remarks\n\nEquivalently, $C_0^{(\\lambda)} = 1$, $C_1^{(\\lambda)} = 2\\lambda x$ and $(n + 1) C_{n+1}^{(\\lambda)} = 2(n + \\lambda) x \\, C_n^{(\\lambda)} - (n + 2\\lambda - 1) C_{n-1}^{(\\lambda)}$ for $n \\ge 1$.\nExample: $C_2^{(\\lambda)}(x) = 2\\lambda(\\lambda + 1) x^2 - \\lambda$.\nCheck: with $u = 2xt - t^2$, the expansion $(1 - u)^{-\\lambda} = 1 + \\lambda u + \\frac{\\lambda(\\lambda + 1)}{2} u^2 + \\cdots$ has $t^2$-coefficient $-\\lambda + 2\\lambda(\\lambda + 1) x^2$.\nThey are orthogonal: $\\int_{-1}^{1} C_m^{(\\lambda)}(x) C_n^{(\\lambda)}(x) (1 - x^2)^{\\lambda - 1/2} \\, dx = 0$ for $m \\ne n$.\nSpecial cases: $C_n^{(1/2)}$ is the [[Legendre polynomial|QFYGJVA996]] $P_n$, and $C_n^{(1)}$ is the [[Chebyshev polynomial of the second kind|7CATGEN6KH]] $U_n$.\nNon-example: $\\lambda = 0$ is excluded, because the generating function is then the constant $1$ and gives $C_n^{(0)} = 0$ for all $n \\ge 1$.\n",
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