orthogonality of Chebyshev polynomials

By expanding the of de Moivre identityMathworldPlanetmath


to sum, one obtains as real partDlmfMathworldPlanetmath certain terms containing power products of cosφ and sinφ, the latter ones only with even exponentsMathworldPlanetmath.  When these are expressed with cosines (sin2φ=1-cos2φ), the real part becomes a polynomialPlanetmathPlanetmath Tn of degree n in the argument (http://planetmath.org/Argument2) cosφ:

cosnφ=Tn(cosφ) (1)

This can be written equivalently (http://planetmath.org/Equivalent3)

Tn(x)=cos(narccosx). (2)

It’s a question of Chebyshev polynomial of first kindDlmfMathworldPlanetmath and of n (cf. special cases of hypergeometric function).

For showing the orthogonality of Tm and Tn we start from the integralDlmfPlanetmathPlanetmath 0πcosmφcosnφdφ, which via the substitution


changes to

0πcosmφcosnφdφ=-1-1Tm(x)Tn(x)dx1-x2. (3)

The left of this equation is evaluated by using the product formula in the entry trigonometric identities:

0πcosmφcosnφdφ=120π(cos(m-n)φ+cos(m+n)φ)𝑑φ={0 for mn,π2 for m=n0.

By (3), we thus have

-11Tm(x)Tn(x)dx1-x2={0 for mn,π2 for m=n0,

which means the orthogonality of the polynomials Tm(x) and Tn(x) weighted by 11-x2.

Any Riemann integrablePlanetmathPlanetmath real function f, defined on  [-1, 1],  may be expanded to the series



aj=2π-11f(x)Tj(x)dx1-x2  (j=0, 1, 2,)

This concerns especially the polynomials  f(x):=xn,  for which we obtain

xn =cosnφ=coshniφ= 2-n(eiφ+e-iφ)
= 2-n[(n0)(eniφ+e-niφ)+(n1)(e(n-2)iφ+e-(n-2)iφ)+]

(If n is even, the last term contains T0(x) but its coefficient is only a half of the middle number of the Pascal’s triangle row in question.)  Explicitly:

x2= 2-1(T2+T0)
x3= 2-2(T3+3T1)
x4= 2-3(T4+4T2+3T0)
x5= 2-4(T5+5T3+10T1)
x6= 2-5(T6+6T4+15T2+10T0)
x7= 2-6(T7+7T5+21T3+35T1)
x8= 2-7(T8+8T6+28T4+56T2+36T0)
x9= 2-8(T9+9T7+36T5+84T3+126T1)


  • 1 Pentti Laasonen: Matemaattisia erikoisfunktioita.  Handout No. 261. Teknillisen Korkeakoulun Ylioppilaskunta; Otaniemi, Finland (1969).
Title orthogonality of Chebyshev polynomials
Canonical name OrthogonalityOfChebyshevPolynomials
Date of creation 2013-03-22 18:54:42
Last modified on 2013-03-22 18:54:42
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Derivation
Classification msc 33C45
Classification msc 33D45
Classification msc 42C05
Related topic OrthogonalPolynomials
Related topic LaguerrePolynomial
Related topic ChangeOfVariableInDefiniteIntegral
Related topic DeterminationOfFourierCoefficients
Related topic OrthogonalityOfLegendrePolynomials
Related topic PropertiesOfOrthogonalPolynomials