Particular Values of Chebyshev Polynomials of the Second Kind

Theorem

Chebyshev Polynomials of the Second Kind of $1$

$\map {U_n} 1 = n + 1$

Chebyshev Polynomials of the Second Kind of $-1$

$\map {U_n} {-1} = \paren {-1}^n \paren {n + 1}$

Chebyshev Polynomial of the Second Kind of $-x$

$\map {U_n} {-x} = \paren {-1}^n \map {U_n} x$

Chebyshev Polynomials of the Second Kind of $0$

$\map {U_n} 0 = \begin {cases} \paren {-1}^{n / 2} & : \text {$n$ even} \\ 0 & : \text {$n$ odd} \end {cases}$