Chebyshev differential equation

(1 - z^{2}) w^{''} - z w^{'} + n^{2} w = 0

is obtained from the harmonic equation

w_{xx} + n^{2} w = 0

z = cos x

Chebyshev derivative Sturm-Liouville pair

The Chebyshev equation has self-adjoint form

(1 - x^{2} T_{n}^{'})^{'} + \frac{n ^{2}}{1 - x ^{2}} T_{n} = 0.

For

U_{n} = T_{n}^{'}

, differentiation gives

((1 - x^{2})^{3/2} U_{n}^{'})^{'} + (n^{2} - 1) 1 - x^{2} U_{n} = 0.

The respective orthogonality weights are

(1 - x^{2})^{- 1/2}

and

(1 - x^{2})^{1/2}

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