Mathematical Methods for Physics and Engineering - Matematica.NET

18.9 HERMITE FUNCTIONS105−1.5H 0H 1−1−0.5H 2H 3x0.5 1 1.5−5−10Figure 18.8The first four Hermite polynomials.a n+2 = a n+4 = ···= 0, and so one solution of Hermite’s equation is a polynomialof order n. For even n, it is conventional to choose a 0 =(−1) n/2 n!/(n/2)!, whereasfor odd n one takes a 1 =(−1) (n−1)/2 2n!/[ 1 2(n − 1)]!. These choices allow a generalsolution to be written asH n (x) =(2x) n − n(n − 1)(2x) n−1 +[n/2]∑=m=0n(n − 1)(n − 2)(n − 3)(2x) n−4 − ···(18.128)2!(−1) m n!m!(n − 2m)! (2x)n−2m , (18.129)where H n (x) is called the nth Hermite polynomial and the notation [n/2] denotesthe integer part of n/2. We note in particular that H n (−x) =(−1) n H n (x). Thefirst few Hermite polynomials are given byH 0 (x) =1, H 3 (x) =8x 2 − 12x,H 1 (x) =2x, H 4 (x) =16x 4 − 48x 2 +12,H 2 (x) =4x 2 − 2, H 5 (x) =32x 5 − 160x 3 + 120x.The functions H 0 (x), H 1 (x), H 2 (x) andH 3 (x) are plotted in figure 18.8.625