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18 Polynomial Approximation of Differential Equations
Together with (1.7.6), we obtain a very simple relation to evaluate the derivatives, i.e.
(1.7.8) H ′ n(x) = 2nHn−1(x), n ≥ 1, x ∈ R.
We give the plots of Hermite polynomials in figures 1.7.1 and 1.7.2. The sizes of
the windows are respectively [−5,5] × [−900,900] and [−5,5] × [−450000,450000].
Figure 1.7.1 - Hermite polynomials Figure 1.7.2 - The ninth Hermite
for 1 ≤ n ≤ 6. polynomial.
It is worthwhile to mention the following result.
Theorem 1.7.1 - For any n ≥ 4, the successive values of the relative maxima of
|Hn(x)| are increasing for x ≥ 0.
Other asymptotic properties can be found in szegö(1939).
The Hermite polynomials can be expressed in term of the Laguerre polynomials accord-
ing to