Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONS4U 2U 32U 0U 1−1 −0.5 0.5 1−2−4Figure 18.4The first four Chebyshev polynomials of the second kind.special values:T n (1) = 1, T n (−1) = (−1) n , T 2n (0) = (−1) n , T 2n+1 (0) = 0.The first few Chebyshev polynomials of the second kind are also easily foundand readU 0 (x) =1,U 1 (x) =2x,U 2 (x) =4x 2 − 1, U 3 (x) =8x 3 − 4x,U 4 (x) =16x 4 − 12x 2 +1, U 5 (x) =32x 5 − 32x 3 +6x.The functions U 0 (x), U 1 (x), U 2 (x) and U 3 (x) are plotted in figure 18.4. TheChebyshev polynomials U n (x) also satisfy U n (−x) =(−1) n U n (x), which may bededuced from (18.57) and (18.58), and have the special values:U n (1) = n +1, U n (−1) = (−1) n (n +1), U 2n (0) = (−1) n , U 2n+1 (0) = 0.◮Show that the Chebyshev polynomials U n (x) satisfy the differential equation(1 − x 2 )U n ′′ (x) − 3xU n(x)+n(n ′ +2)U n (x) =0. (18.59)From (18.58), we have V n+1 =(1− x 2 ) 1/2 U n and these functions satisfy the Chebyshevequation (18.54) with ν = n +1,namely(1 − x 2 )V n+1 ′′ − xV n+1 ′ +(n +1) 2 V n+1 =0. (18.60)598