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  "body": "\n## Definition\n\nThe *Chebyshev polynomials of the second kind* are the polynomials $U_n \\in \\Z[x]$, $n \\ge 0$, in the [[polynomial ring|0SDC934R52]] $\\Z[x]$ given by\n$$U_0 = 1, \\qquad U_1 = 2x, \\qquad U_{n+1} = 2x \\, U_n - U_{n-1} \\quad (n \\ge 1).$$\n\n## Remarks\n\nEquivalently, $U_n$ is the unique polynomial with $U_n(\\cos \\theta) \\sin \\theta = \\sin((n + 1)\\theta)$ for all real $\\theta$.\nExample: $U_2 = 2x \\cdot 2x - 1 = 4x^2 - 1$, and indeed $\\sin(3\\theta) = \\sin \\theta \\, (3 - 4 \\sin^2 \\theta) = \\sin \\theta \\, (4 \\cos^2 \\theta - 1)$.\nNon-example: the [[Chebyshev polynomials|M6EAR1JHPP]] of the first kind $T_n$ satisfy the same recurrence but start with $T_1 = x$, so they fail the clause $U_1 = 2x$.\nThe polynomial $U_n$ has degree $n$, leading coefficient $2^n$, and roots $\\cos \\frac{k\\pi}{n + 1}$ for $k = 1, \\dots, n$.\nThey are orthogonal: $\\int_{-1}^{1} U_m(x) U_n(x) \\sqrt{1 - x^2} \\, dx = \\frac{\\pi}{2} \\delta_{mn}$.\nThey are linked to the first kind by $T_n' = n \\, U_{n-1}$ for $n \\ge 1$.\n",
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