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 between, relation and/or kinds:

    “There is a certain standard of grace and beauty which consists in a certain relation between our nature, such as it is, weak or strong, and the thing which pleases us. Whatever is formed according to this standard pleases us, be it house, song, discourse, verse, prose, woman, birds, rivers, trees, room, dress, and so on. Whatever is not made according to this standard displeases those who have good taste.”
    —Blaise Pascal (1623–1662)

    “Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
    —William Harmon (b. 1938)

    “If you give me your attention, I will tell you what I am:
    I’m a genuine philanthropist—all other kinds are sham.
    Each little fault of temper and each social defect
    In my erring fellow creatures, I endeavor to correct.”
    —Sir William Schwenck Gilbert (1836–1911)