Page 328 - Elements of Distribution Theory

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052184472Xc10 CUNY148/Severini May 24, 2005 2:50

314 Orthogonal Polynomials

Corollary 10.1. For each n = 0, 1,..., let H n denote the function defined by (10.9). Then

H n+1 (x) = xH n (x) − H (x), n = 0, 1, 2,....
n
Starting with H 0 (x) = 1, it is straightforward to use Corollary 10.1 to determine the first
several Hermite polynomials. The results are given in the following corollary; the proof is
left as an exercise.

Corollary 10.2. Let H n denote the nth Hermite polynomial, n = 1, 2, 3, 4. Then

2
H 1 (x) = x, H 2 (x) = x − 1,
3
4
2
H 3 (x) = x − 3x, H 4 (x) = x − 6x + 3.
Equation (10.9) can be used to find
∞

2
H n (x) φ(x) dx.
−∞
The result is given in the following corollary; the proof is left as an exercise.
Corollary 10.3. For each n = 1, 2,..., let H n denote the nth Hermite polynomial. Then
∞

2
H n (x) φ(x) dx = n!.
−∞
Using the expression (10.9) for H n ,itis straightforward to calculate integrals of the form

x
H n (t)φ(t) dt.
−∞
Theorem 10.8. Let H n denote the nth Hermite polynomial. Then
x
H n (t)φ(t) dt =−H n−1 (x)φ(x).
−∞
Proof. Note that

x x d n
H n (t)φ(t) dt = (−1) n φ(t) dt
dt n
−∞ −∞
d n−1 x x
n
= (−1) φ(t) =−H n−1 (t)φ(t) =−H n−1 (x)φ(x),
dt n−1 −∞ −∞
proving the result.
Hence, any integral of the form
x
f (t)φ(t) dt,
−∞
where f is a polynomial, can be integrated exactly in terms of H 0 , H 1 ,..., φ, and the
standard normal distribution function .