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 U_n (the polynomial of degree n) with U_n+1 instead.

Turán's inequalities for the Chebyshev polynomials are

and

Read more about this topic: Chebyshev Polynomials

Famous quotes containing the words relation between, relation and/or kinds:

“We shall never resolve the enigma of the relation between the negative foundations of greatness and that greatness itself.”
—Jean Baudrillard (b. 1929)

“Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.”
—Henry David Thoreau (1817–1862)

“There are two kinds of liars the kind that lie and thekind that don’t lie the kind that lie are no good.”
—Gertrude Stein (1874–1946)