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  "body": "\n## Summary\n\nThe Chebyshev polynomials of the second kind $U_n$ are the polynomials with $U_n(\\cos \\theta) \\sin \\theta = \\sin((n + 1)\\theta)$.\nThey are the companions of the [[Chebyshev polynomials|M6EAR1JHPP]] of the first kind $T_n$, and satisfy the same three-term recurrence with a different start.\nThey are orthogonal on $[-1, 1]$ for the weight $\\sqrt{1 - x^2}$, and they are the [[Gegenbauer polynomials|5TRFA6F79B]] with parameter $1$.\n\n## Explanations\n\n### L1 human\n\nThe ratio $\\sin((n + 1)\\theta) / \\sin \\theta$ can always be written as a polynomial in $\\cos \\theta$.\nThat polynomial is $U_n$.\nFor example, $\\sin(3\\theta) / \\sin \\theta = 4 \\cos^2 \\theta - 1$, so $U_2(x) = 4x^2 - 1$.\nThey show up as derivatives of the first-kind polynomials, as determinants of tridiagonal matrices, and in the semicircle law of random matrices.\n",
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