Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.4 CHEBYSHEV FUNCTIONS

in which the coefficientsanare given by

an=

2

π

∫ 1

− 1

f(x)Un(x)(1−x^2 )^1 /^2 dx.

Generating functions

The generating functions for the Chebyshev polynomials of the first and second

kinds are given, respectively, by

GI(x, h)=

1 −xh

1 − 2 xh+h^2

=

∑∞

n=0

Tn(x)hn, (18.63)

GII(x, h)=

1

1 − 2 xh+h^2

=

∑∞

n=0

Un(x)hn. (18.64)

These prescriptions may be proved in a manner similar to that used in sec-

tion 18.1.2 for the generating function of the Legendre polynomials. For the

Chebyshev polynomials, however, the generating functions are of less practical

use, since most of the useful results can be obtained more easily by taking

advantage of the trigonometric forms (18.55), as illustrated below.

Recurrence relations

There exist many useful recurrence relationships for the Chebyshev polynomials

Tn(x)andUn(x). They are most easily derived by settingx=cosθand using

(18.55) and (18.58) to write

Tn(x)=Tn(cosθ)=cosnθ, (18.65)

Un(x)=Un(cosθ)=

sin(n+1)θ

sinθ

. (18.66)

One may then use standard formulae for the trigonometric functions to derive

a wide variety of recurrence relations. Of particular use are the trigonometric

identities

cos(n±1)θ=cosnθcosθ∓sinnθsinθ, (18.67)

sin(n±1)θ=sinnθcosθ±cosnθsinθ. (18.68)

Show that the Chebyshev polynomials satisfy the recurrence relations

Tn+1(x)− 2 xTn(x)+Tn− 1 (x)=0, (18.69)

Un+1(x)− 2 xUn(x)+Un− 1 (x)=0. (18.70)

Adding the result (18.67) with the plus sign to the corresponding result with a minus sign
gives

cos(n+1)θ+cos(n−1)θ=2cosnθcosθ.