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12 Polynomial Approximation of Differential Equations
These are also ultraspherical polynomials. By virtue of (1.3.6) and (1.5.1), we have
2 (1.5.13) Un = δn+1 P n , n ∈ N.
( 1
, 1
2 )
The Un’s satisfy similar properties to those of Chebyshev polynomials of the first kind.
1.6 Laguerre polynomials
In this section, we introduce another family of polynomial solutions of the Sturm-
Liouville problem. Let α > −1 and I =]0,+∞[, then define the coefficients in (1.1.1)
to be
a(x) = x α+1 e −x , ∀x ∈ Ī,
b(x) = 0, w(x) = x α e −x , ∀x ∈ I.
This choice leads to the eigenvalue problem
(1.6.1) xu ′′ + (α + 1 − x) u ′ + λu = 0.
This admits polynomial solutions only if λ = n, n ∈ N. Therefore, we define the
n-degree Laguerre polynonial L (α)
n
(1.6.1), satisfying the condition
(1.6.2) L (α)
n (0) :=
We have the Rodrigues’ formula
(1.6.3) e −x x α L (α)
n (x) = 1
n!
The following expression also holds:
(1.6.4) L (α)
n (x) =
n
k=0
to be the unique solution of the singular problem
n + α
n
, n ∈ N, α > −1.
dn dxn −x n+α
e x , n ∈ N, x ∈ I.
k n + α (−1)
n − k k !
x k , n ∈ N, x ∈ Ī.