2316 CHAPTER 68. A DIFFERENT KIND OF STOCHASTIC INTEGRATION
Now also
exp(
t (−x)− t2
2
)=
∞
∑n=0
Hn (−x) tn
and taking successive derivatives with respect to t of the left side and evaluating at t = 0yields
Hn (−x) = (−1)n Hn (x) .
Summarizing these as in [102],
H ′n (x) = Hn−1 (x) ,n≥ 1, H0 (x) = 0,H1 (x) = xxHn (x)−Hn−1 (x) = (n+1)Hn+1 (x) , n≥ 1
Hn (−x) = (−1)n Hn (x)(68.1.3)
Clearly, these relations show that all of these Hn are polynomials. Also the degree of Hn (x)is n and the coefficient of xn is 1/n!.
Definition 68.1.1 You can also consider Hermite polynomials which depend on λ . Theseare defined as follows:
Hn (x,λ )≡(−λ )n
n!e
12λ
x2 ∂ n
∂xn
(e−
12λ
x2)
You can see clearly that these are polynomials in x. For example, let n = 2. Then youwould have from the above definition.
H0 (x,λ ) = 1, H1 (x,λ ) =(−λ )1
1!e
12λ
x2 ∂
∂x
(e−
12
x2λ
)= x
H2 (x,λ )≡(−λ )2
2!e
12λ
x2 ∂ 2
∂x2
(e−
12λ
x2)=
12
x2− 12
λ
The idea is you end up with polynomials of degree n times e−x2/2λ in the derivative partand then this cancels with ex2/2λ to leave you with a polynomial of degree n. Also theleading term will always be xn
n! which is easily seen from the above. Then there are somerelationships satisfied by these.
Say n > 1 in what follows.
∂
∂xHn (x,λ ) =
xλ
(−λ )n e12
x2λ
n!∂ n
∂xn
(e−
12λ
x2)+
(−λ )n
n!e
12λ
x2 ∂ n
∂xn
(∂
∂xe−
12λ
x2)
=xλ
(−λ )n e12
x2λ
n!∂ n
∂xn
(e−
12λ
x2)+
(−λ )n
n!e
12λ
x2 ∂ n
∂xn
(− x
λe−
12
x2λ
)Now since n > 1, that last term reduces to
(−λ )n
n!e
12λ
x2[− x
λ
∂ n
∂xn
(e−
12
x2λ
)+n
∂
∂x
(− x
λ
)∂ n−1
∂xn−1
(e−
12
x2λ
)]