SPECIAL FUNCTIONS
U 0
U 1
U 2
U 3
− (^1) − 0. 5 0. 5 1
− 2
2
− 4
4
Figure 18.4 The first four Chebyshev polynomials of the second kind.special values:
Tn(1) = 1,Tn(−1) = (−1)n,T 2 n(0) = (−1)n,T 2 n+1(0) = 0.The first few Chebyshev polynomials of the second kind are also easily foundand read
U 0 (x)=1, U 1 (x)=2x,U 2 (x)=4x^2 −1, U 3 (x)=8x^3 − 4 x,U 4 (x)=16x^4 − 12 x^2 +1, U 5 (x)=32x^5 − 32 x^3 +6x.The functionsU 0 (x),U 1 (x),U 2 (x)andU 3 (x) are plotted in figure 18.4. The
Chebyshev polynomialsUn(x) also satisfyUn(−x)=(−1)nUn(x), which may be
deduced from (18.57) and (18.58), and have the special values:
Un(1) =n+1,Un(−1) = (−1)n(n+1),U 2 n(0) = (−1)n,U 2 n+1(0) = 0.
Show that the Chebyshev polynomialsUn(x)satisfy the differential equation
(1−x^2 )U′′n(x)− 3 xUn′(x)+n(n+2)Un(x)=0. (18.59)From (18.58), we haveVn+1=(1−x^2 )^1 /^2 Unand these functions satisfy the Chebyshev
equation (18.54) withν=n+1, namely
(1−x^2 )Vn′′+1−xVn′+1+(n+1)^2 Vn+1=0. (18.60)