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Special Families of Polynomials 19
(1.7.9) Hn(x) = (−1) n/2 2 n (n/2)! L (−1/2)
n/2 (x 2 ), if n is even,
(1.7.10) Hn(x) = (−1) (n−1)/2 2 n ((n − 1)/2)! x L (1/2)
(n−1)/2 (x2 ), if n is odd.
The formulas above are easily checked with the help of (1.6.1)-(1.6.2) and (1.7.1)-(1.7.2)-
(1.7.3).
Inspired by (1.7.9) and (1.7.10), recalling the definition (1.6.12)-(1.6.13), it is nat-
ural to define the scaled Hermite functions by
(1.7.11)
(1.7.12)
Consequently
ˆ Hn(x) := ˆ L (−1/2)
n/2 (x 2 ), if n is even,
ˆ Hn(x) := x ˆ L (1/2)
(n−1)/2 (x2 ), if n is odd.
(1.7.13) ˆ Hn(0) = 1, if n is even,
(1.7.14) ˆ H ′ n(0) = 1, if n is odd.
Derivatives can be computed via (1.6.14) and (1.6.15).
Finally we show the plots of ˆ Hn, 1 ≤ n ≤ 9, in figure 1.7.3. The window measures
[−5,5] × [−3,3].