Orthogonality of Chebyshev Polynomials of the Second Kind

Theorem

The Chebyshev polynomials of the second kind form a set of orthogonal polynomials with respect to:

the closed real interval $\closedint {-1} 1$

the weight function $\map w x := \sqrt {1 - x^2}$ on $\closedint {-1} 1$

That is:

Inequality

$\ds \int_{-1}^1 \sqrt {1 - x^2} \, \map {U_m} x \map {U_n} x \rd x = 0$

when $m \ne n$.

Equality

$\ds \int_{-1}^1 \sqrt {1 - x^2} \, \paren {\map {U_n} x}^2 \rd x = \dfrac \pi 2$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 30$: Chebyshev Polynomials: Orthogonality: $30.38$, $30.39$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 31$: Chebyshev Polynomials: Orthogonality: $31.38.$, $31.39.$