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  "body": "\n## Summary\n\nThe Jacobi polynomials $P_n^{(\\alpha, \\beta)}$ are the [[classical orthogonal polynomials|47WJ34NC9K]] on $[-1, 1]$ for the weight $(1 - x)^\\alpha (1 + x)^\\beta$, with $\\alpha, \\beta \u003e -1$.\nThey include the [[Gegenbauer polynomials|5TRFA6F79B]], the [[Legendre polynomials|QFYGJVA996]] and both kinds of Chebyshev polynomials as special cases, up to constant factors.\nThey are terminating [[hypergeometric functions|463X5V31D9]], and they appear in harmonic analysis on spheres and on compact symmetric spaces.\n\n## Explanations\n\n### L1 human\n\nThe weight $(1 - x)^\\alpha (1 + x)^\\beta$ on the interval $[-1, 1]$ can favour one end, the other end, or both, depending on the two exponents.\nFor each choice there is one family of polynomials, perpendicular to each other for this weight: the Jacobi polynomials.\nWith $\\alpha = \\beta = 0$ the weight is constant and the family is the Legendre polynomials.\nFor example, $P_1^{(0,0)}(x) = x$ is perpendicular to the constant $1$ because $\\int_{-1}^1 x \\, dx = 0$.\n",
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