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

Read more about this topic:  Chebyshev Polynomials

Famous quotes containing the words relation and/or kinds:

    “Skepticism is unbelief in cause and effect. A man does not see, that, as he eats, so he thinks: as he deals, so he is, and so he appears; he does not see that his son is the son of his thoughts and of his actions; that fortunes are not exceptions but fruits; that relation and connection are not somewhere and sometimes, but everywhere and always; no miscellany, no exemption, no anomaly,—but method, and an even web; and what comes out, that was put in.”
    —Ralph Waldo Emerson (1803–1882)

    “We have two kinds of “conference.” One is that to which the office boy refers when he tells the applicant for a job that Mr. Blevitch is “in conference.” This means that Mr. Blevitch is in good health and reading the paper, but otherwise unoccupied. The other type of “conference” is bona fide in so far as it implies that three or four men are talking together in one room, and don’t want to be disturbed.”
    —Robert Benchley (1889–1945)