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10.6 Suggestions for Further Reading 321

10.15 Show that the Hermite polynomials satisfy
∞

H n (x) n 2
z = exp(xz − z /2), x ∈ R, z ∈ R.
n!
n=0
Using this result, give a relationship between H n (0), n = 0, 1,..., and the moments of the
standard normal distribution.
10.16 Let L n denote the nth Laguerre polynomial. Show that
∞

2
L n (x) exp(−x) dx = 1.
0
10.17 Let p 0 , p 1 , p 2 denote the orthogonal polynomials found in Exercise 10.3. Find the zeros x 1 , x 2
of p 2 and the constants λ 1 ,λ 2 such that
∞

1
f ( j) = λ 1 f (x 1 ) + λ 2 f (x 2 )
2 j+1
j=0
for all polynomials f of degree 3 or less.

10.6 Suggestions for Further Reading

Orthogonal polynomials are a classical topic in mathematics. Standard references include
Freud (1971), Jackson (1941), and Szeg¨o (1975); see also Andrews et al. (1999, Chapters 5–7)
and Temme (1996, Chapter 6). Many useful properties of the classical orthogonal polynomials are
given in Erd´elyi (1953b, Chapter X).
Gaussian quadrature is discussed in many books covering numerical integration; see, for example,
Davis and Rabinowitz (1984). Evans and Swartz (2000, Chapter 5) and Thisted (1988, Chapter 5)
discuss Gaussian quadrature with particular emphasis on statistical applications. Tables listing the
constants needed to implement these methods are available in Abramowitz and Stegun (1964) and
Stroud and Secrest (1966).