Untitled - Cdm.unimo.it

126 Polynomial Approximation of Differential Equations
with un = P (ν,ν)
n , ν > −1 and ν = −1 2 .
After tedious calculation, we recover from (7.1.2) the expression of the derivatives of
the ultraspherical polynomials, i.e.
(7.1.3)
d
dx un =
×
[(n−1)/2]
m=0
Γ(n + ν + 1)
Γ(n + 2ν + 1)
2n − 4m + 2ν − 1
n − 2m + 2ν
Γ(n − 2m + 2ν + 1)
Γ(n − 2m + ν)
un−2m−1
for any n ≥ 1 and ν > −1 with ν = −1 2 . The symbol [•] denotes the integer part of •.
Particularly interesting is the Legendre case (ν = 0), where (7.1.3) takes the form
(7.1.4)
d
dx Pn =
[(n−1)/2]
m=0
(2n − 4m − 1)Pn−2m−1, n ≥ 1.
Using the same arguments, recalling (1.5.10), we get in the Chebyshev case
(7.1.5)
d
dx Tn =
⎧
⎪⎨
T0
n/2−1
2n
m=0
⎪⎩ nT0 + 2n
T2m+1
(n−1)/2
m=1
T2m
if n = 1,
if n ≥ 2 is even,
if n ≥ 3 is odd.
Substituting formula (7.1.3) into (7.1.1), we can relate the coefficients of p ′ to those of
p. First of all, we note that c (1)
(7.1.6) c (1)
i
2i + 2ν + 1
=
i + 2ν + 1
n = 0. Then, for ν > −1 with ν = −1 2
Γ(i + 2ν + 2)
Γ(i + ν + 1)
n
j=i+1
i+j odd
,
, we have
Γ(j + ν + 1)
Γ(j + 2ν + 1) cj,
0 ≤ i ≤ n − 1.
In the Legendre case (ν = 0), the above formula yields (see also the appendix in the
book of gottlieb and orszag(1977))