18.9 HERMITE FUNCTIONS
H 0
H 1
H 2
H 3
− 1. 5 − 1 − 0. 5 0.^511.^5
5
10
− 5
− 10
xFigure 18.8 The first four Hermite polynomials.an+2=an+4=···= 0, and so one solution of Hermite’s equation is a polynomial
of ordern. For evenn, it is conventional to choosea 0 =(−1)n/^2 n!/(n/2)!, whereas
for oddnone takesa 1 =(−1)(n−1)/^22 n!/[^12 (n−1)]!. These choices allow a general
solution to be written as
Hn(x)=(2x)n−n(n−1)(2x)n−^1 +n(n−1)(n−2)(n−3)
2!(2x)n−^4 −···(18.128)=[∑n/2]m=0(−1)mn!
m!(n− 2 m)!(2x)n−^2 m, (18.129)whereHn(x) is called thenthHermite polynomialand the notation [n/2] denotes
the integer part ofn/2. We note in particular thatHn(−x)=(−1)nHn(x). The
first few Hermite polynomials are given by
H 0 (x)=1,H 3 (x)=8x^2 − 12 x,H 1 (x)=2x, H 4 (x)=16x^4 − 48 x^2 +12,H 2 (x)=4x^2 − 2 ,H 5 (x)=32x^5 − 160 x^3 + 120x.The functionsH 0 (x),H 1 (x),H 2 (x)andH 3 (x) are plotted in figure 18.8.