Orthogonality of Chebyshev Polynomials of the Second Kind/Inequality

Theorem

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

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

when $m \ne n$.

Proof

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Sources

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