Untitled - Cdm.unimo.it

Orthogonality 33
only mention the relation existing between some Jacobi polynomials and Chebyshev
polynomials, namely
(2.6.1) P (α,α)
n
×
k=0
=
Γ(2α + 1) Γ(n + α + 1)
Γ(α + 1) Γ(n + 2α + 1)
n Γ(k + α + 1
2
1 Γ(α + 2 ) 2
k! (n − k)!
) Γ(n − k + α + 1
2 )
T |n−2k|
, n ∈ N, α > − 1
2 .
A problem related to the previous one, is the determination of the weighted norm
of a polynomial, when the expansion coefficients, corresponding to a basis orthogonal
with respect to another weight function, are known. We give an example. Sometimes,
in practical applications it is useful to evaluate the integral 1
−1 p2 dx, p ∈ Pn. If
p is known in terms of its Legendre coefficients, (2.3.6) is the formula we are looking
for. The situation becomes more complicated when p is given in terms of the Fourier
coefficients of another basis, say the Chebyshev basis. Let us assume that
(2.6.2) p =
Then
(2.6.3)
1
−1
p 2 dx =
n
k=0
ckTk.
The help of some trigonometry leads to the formula
(2.6.4) Ikj =
n n
ckcjIkj,
1
where Ikj = TkTj dx.
k=0 j=0
−1
π
0
cos kθ cos jθ sinθ dθ
⎧
⎪⎨
0 if k + j is odd,
=
⎪⎩
1 1
+
1 − (k + j) 2 1 − (k − j) 2 if k + j is even.