The Chebyshev polynomials of the first kind are defined by:
Similarly, the Chebyshev polynomials of the second kind are defined by:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by . Hence and .
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below.
The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval[−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.[1]
These polynomials were named after Pafnuty Chebyshev.[3] The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
Definitions
Recurrence definition
The Chebyshev polynomials of the first kind are obtained from the recurrence relation:
The recurrence also allows to represent them explicitly as the determinant of a tridiagonal matrix of size :
That cos nx is an nth-degree polynomial in cos x can be seen by observing that cos nx is the real part of one side of de Moivre's formula:
The real part of the other side is a polynomial in cos x and sin x, in which all powers of sin x are even and thus replaceable through the identity cos2x + sin2x = 1.
By the same reasoning, sin nx is the imaginary part of the polynomial, in which all powers of sin x are odd and thus, if one factor of sin x is factored out, the remaining factors can be replaced to create a (n−1)st-degree polynomial in cos x.
Commuting polynomials definition
Chebyshev polynomials can also be characterized by the following theorem:[5]
If is a family of monic polynomials with coefficients in a field of characteristic such that and for all
and , then, up to a simple change of variables, either for all or
for all .
Pell equation definition
The Chebyshev polynomials can also be defined as the solutions to the Pell equation:
in a ringR[x].[6] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:
Relations between the two kinds of Chebyshev polynomials
The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequencesṼn(P, Q) and Ũn(P, Q) with parameters P = 2x and Q = 1:
It follows that they also satisfy a pair of mutual recurrence equations:[7]
The second of these may be rearranged using the recurrence definition for the Chebyshev polynomials of the second kind to give:
Using this formula iteratively gives the sum formula:
while replacing and using the derivative formula for gives the recurrence relationship for the derivative of :