Particular Values of Chebyshev Polynomials of the Second Kind/-1

Theorem

Let $\map {U_n} x$ denote the Chebyshev polynomial of the second kind of order $n$.

Then:

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

Proof

\(\ds \map {U_n} {-x}\)

\(=\)

\(\ds \paren {-1}^n \map {U_n} x\)

Particular Values of Chebyshev Polynomials of the Second Kind: $-x$

\(\ds \leadsto \ \ \)

\(\ds \map {U_n} {-1}\)

\(=\)

\(\ds \paren {-1}^n \map {U_n} 1\)

setting $x = 1$

\(\ds \)

\(=\)

\(\ds \paren {-1}^n \paren {n + 1}\)

Particular Values of Chebyshev Polynomials of the Second Kind: $1$

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 30$: Chebyshev Polynomials: Special Values: $30.34$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 31$: Chebyshev Polynomials: Special Values: $31.34.$