Untitled - Cdm.unimo.it

14 Polynomial Approximation of Differential Equations
Figure 1.6.1 - Laguerre polynomials Figure 1.6.2 - The Laguerre
for α = 0 and 1 ≤ n ≤ 6. polynomial L (0)
7 .
Further insight into the asymptotic behavior can be given. We recall two results
(see szegö(1939), p.171 and p.234). The first one shows the behavior when x → +∞.
In the second one, the case n → +∞ is considered. It is convenient to define Q (α)
n (x) :=
e −x/2 x (α+1)/2 L (α)
n (x).
Theorem 1.6.2 - For any α > −1 and n ≥ 2, the values of the relative maxima of
|Q (α)
n (x)| form an increasing sequence when x ∈]x0,+∞[, where
x0 := 0 if α ≤ 1, x0 :=
α 2 − 1
2n + α + 1
if α > 1.
Theorem 1.6.3 - Let α > −1, then for any γ > 0 and 0 < η < 4 we can find two
positive constants C1, C2 such that
(1.6.10) max
x∈[γ,(4−η)n]
(1.6.11) max
x∈[γ,+∞[
|Q (α)
n (x)| ≈ C1 n α/2 ,
|Q (α)
n (x)| ≈ C2 6√ n n α/2 .