Chebyshev Polynomials - Relation Between Chebyshev Polynomials of The First and Second Kinds

Relation Between Chebyshev Polynomials of The First and Second Kinds

The Chebyshev polynomials of the first and second kind are closely related by the following equations

, where n is odd.
, where n is even.

The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations

This relationship is used in the Chebyshev spectral method of solving differential equations.

Equivalently, the two sequences can also be defined from a pair of mutual recurrence equations:

These can be derived from the trigonometric formulae; for example, if, then

\begin{align} T_{n+1}(x) &= T_{n+1}(\cos(\vartheta)) \\ &= \cos((n + 1)\vartheta) \\ &= \cos(n\vartheta)\cos(\vartheta) - \sin(n\vartheta)\sin(\vartheta) \\ &= T_n(\cos(\vartheta))\cos(\vartheta) - U_{n-1}(\cos(\vartheta))\sin^2(\vartheta) \\ &= xT_n(x) - (1 - x^2)U_{n-1}(x). \\ \end{align}

Note that both these equations and the trigonometric equations take a simpler form if we, like some works, follow the alternate convention of denoting our Un (the polynomial of degree n) with Un+1 instead.

Turán's inequalities for the Chebyshev polynomials are

and

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