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Special Families of Polynomials 17
a(x) = w(x) = e −x2
, b(x) = 0, ∀x ∈ I ≡ R.
Therefore, one easily gets the differential equation
(1.7.1) H ′′
n(x) − 2xH ′ n(x) + 2nHn(x) = 0, n ∈ N, x ∈ R.
The normalizing condition is
(1.7.2) Hn(0) := (−1) n/2 n!
(n/2)!
(1.7.3) H ′ n(0) := (−1) (n−1)/2 (n + 1)!
((n + 1)/2)!
if n is even,
if n is odd.
The n th degree polynomial Hn is an even or odd function according to the parity of n.
As usual, we have the Rodrigues’ formula
(1.7.4) Hn(x) = (−1) n dn x2
e e−x2,
n ∈ N, x ∈ R.
dxn More explicitly, we can write
(1.7.5) Hn(x) = n!
[n/2]
m=0
(−1) m
m!
(2x) n−2m
(n − 2m)!
= 2 n x n − n(n − 1)2 n−2 x n−2 + n(n − 1)(n − 2)(n − 3)2 n−5 x n−4 − · · ·,
The corresponding three-term recursion formula is
(1.7.6) Hn(x) = 2xHn−1(x) − 2(n − 1)Hn−2(x), ∀n ≥ 2,
where H0(x) = 1 and H1(x) = 2x.
By differentiating (1.7.4) we also get
(1.7.7) H ′ n(x) = 2xHn(x) − Hn+1(x), n ∈ N, x ∈ R.
n ∈ N, x ∈ R.