Mathematical Methods for Physics and Engineering - Matematica.NET

18.4 CHEBYSHEV FUNCTIONSin which the coefficients a n are given bya n = 2 π∫ 1−1f(x)U n (x)(1 − x 2 ) 1/2 dx.Generating functionsThe generating functions for the Chebyshev polynomials of the first and secondkinds are given, respectively, by1 − xh∞G I (x, h) =1 − 2xh + h 2 = ∑T n (x)h n , (18.63)n=0∞1G II (x, h) =1 − 2xh + h 2 = ∑U n (x)h n . (18.64)These prescriptions may be proved in a manner similar to that used in section18.1.2 for the generating function of the Legendre polynomials. For theChebyshev polynomials, however, the generating functions are of less practicaluse, since most of the useful results can be obtained more easily by takingadvantage of the trigonometric forms (18.55), as illustrated below.Recurrence relationsThere exist many useful recurrence relationships for the Chebyshev polynomialsT n (x) andU n (x). They are most easily derived by setting x =cosθ and using(18.55) and (18.58) to writeT n (x) =T n (cos θ) =cosnθ, (18.65)sin(n +1)θU n (x) =U n (cos θ) = . (18.66)sin θOne may then use standard formulae for the trigonometric functions to derivea wide variety of recurrence relations. Of particular use are the trigonometricidentitiesn=0cos(n ± 1)θ =cosnθ cos θ ∓ sin nθ sin θ, (18.67)sin(n ± 1)θ =sinnθ cos θ ± cos nθ sin θ. (18.68)◮Show that the Chebyshev polynomials satisfy the recurrence relationsT n+1 (x) − 2xT n (x)+T n−1 (x) =0, (18.69)U n+1 (x) − 2xU n (x)+U n−1 (x) =0. (18.70)Adding the result (18.67) with the plus sign to the corresponding result with a minus signgivescos(n +1)θ +cos(n − 1)θ =2cosnθ cos θ.601