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  "body": "\n## Summary\n\nThe Gegenbauer polynomials $C_n^{(\\lambda)}$, also called ultraspherical polynomials, are orthogonal on $[-1, 1]$ for the weight $(1 - x^2)^{\\lambda - 1/2}$.\nUp to constant factors they are the [[Jacobi polynomials|TRD7DSF2KR]] with $\\alpha = \\beta = \\lambda - \\frac12$.\nThey generalize the [[Legendre polynomials|QFYGJVA996]] ($\\lambda = \\frac12$) and give the [[zonal spherical harmonics|Q5D54R3RGE]] on spheres of every dimension at least $2$.\n\n## Explanations\n\n### L1 human\n\nThe Legendre polynomials are tied to the ordinary sphere in space.\nOn a sphere in a space of another dimension, the same role is played by a related family, the Gegenbauer polynomials.\nThey all come from one formula: expand $(1 - 2xt + t^2)^{-\\lambda}$ in powers of $t$, and the coefficients are the $C_n^{(\\lambda)}(x)$.\nFor $\\lambda = \\frac12$ this is the generating function of the Legendre polynomials.\n",
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