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16 Polynomial Approximation of Differential Equations
It is evident that the scaled Laguerre functions have a milder behavior. We note that
ˆL (α)
n (0) = 1, ∀n ∈ N. More details are given in funaro(1990a). By substitution in
(1.6.5), after simplification, one obtains ∀n ≥ 2 the recursion formula
(1.6.14)
ˆ L (α)
n (x) =
with ˆ L (α)
0 (x) = 1 and ˆ L (α)
1 (x) = 4(α+1−x)
(α+1)(x+4) .
Moreover, for the derivatives one has ∀n ≥ 2
d
(1.6.15)
dx ˆ L (α)
4n
n (x) =
(n + α)(4n + x)
6n + α − 1
−
4n + x ˆ L (α)
4(n − 1)2
n−1 (x) +
4n + x − 4
with d
dx ˆ L (α)
0 (x) = 0 and d
4n
(2n + α − 1 − x)
(n + α)(4n + x)
ˆ L (α)
n−1 (x)
dx ˆ L (α)
1
−
4(n − 1)2
4n + x − 4 ˆ L (α)
n−2 (x)
,
(2n + α − 1 − x) d
dx ˆ L (α)
n−1 (x)
2(4n + x − 2)
(4n + x)(4n + x − 4) ˆ L (α) d
n−2 (x) −
dx ˆ L (α)
n−2 (x)
(x) = − α+5
α+1
4
(x+4) 2 .
Applications will be examined in the following chapters (see sections 3.10 and 7.5).
1.7 Hermite polynomials
The set of Hermite polynomials Hn, n ∈ N, is the last family we consider. Following
the notations adopted in szegö(1939) or courant and hilbert(1953), the Hn’s are
solutions of the non singular Sturm-Liouville problem obtained by setting in (1.1.1):
,