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  "body": "\n## Definition\n\nFix a real number $\\lambda \u003e -1/2$ with $\\lambda \\ne 0$, and let $w(x) = (1 - x^2)^{\\lambda - 1/2}$ for $-1 \u003c x \u003c 1$.\nLet $C_n^{\\lambda}$ be the [[Gegenbauer polynomial|5TRFA6F79B]] of degree $n$.\nLet $F \\colon (-1, 1) \\to \\C$ be a function such that $w F$ is integrable on $(-1, 1)$.\nThe *Gegenbauer transform* of $F$ is the sequence\n$$f(n) = \\int_{-1}^{1} w(x)\\, C_n^{\\lambda}(x)\\, F(x)\\, dx, \\qquad n = 0, 1, 2, \\dots$$\n\n## Remarks\n\nThe polynomials $C_n^\\lambda$ are orthogonal for $w$, and $F \\in L^2(w)$ is recovered as $F = \\sum_{n \\ge 0} \\frac{f(n)}{h_n} C_n^{\\lambda}$ with $h_n = \\int_{-1}^1 w\\, (C_n^\\lambda)^2\\, dx$.\nIf $F$ is smooth enough and the boundary terms vanish, the transform of $w^{-1} \\frac{d}{dx}\\left( (1 - x^2)^{\\lambda + 1/2} F' \\right)$ is $-n(n + 2\\lambda) f(n)$.\n\nExample: $\\lambda = 1/2$ gives $w = 1$ and $C_n^{1/2} = P_n$, the Legendre polynomials.\nFor $F(x) = x$ the transform is $f(1) = \\int_{-1}^1 x^2\\, dx = 2/3$ and $f(n) = 0$ for $n \\ne 1$, since $x = P_1(x)$.\n\nNon-example: for $\\lambda = -1/2$ the weight $(1 - x^2)^{-1}$ is not integrable near $\\pm 1$, so $F = 1$ has no transform.\n\nThe value $\\lambda = 0$ is excluded because $C_n^0 = 0$ for $n \\ge 1$ in the usual normalization; the Chebyshev polynomials of the first kind take its place.\n",
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