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Special Families of Polynomials 15
In order to allow more flexible numerical computations, it is useful to introduce
some suitable scaling functions S (α)
n : Ī → R and to define
(1.6.12)
ˆ L (α)
n := S (α)
n L (α)
n , n ∈ N, α > −1.
The aim is to avoid ill-conditioned operations when evaluating point values of Laguerre
polynomials. According to funaro(1990a) an effective choice of S (α)
n is
(1.6.13) S (α)
0 (x) := 1, S (α)
n (x) :=
n n
+ α
n
k=1
1 + x
4k
−1 , n ≥ 1.
We will refer to ˆ L (α)
n , n ∈ N, as the family of scaled Laguerre functions. It is clear that
these are not polynomials. Plots of ˆ L (0)
n , 1 ≤ n ≤ 12, are given in figure 1.6.3. Now
the window size is [0,50] × [−300,300].
Figure 1.6.3 - Scaled Laguerre functions for α = 0 and 1 ≤ n ≤ 12.