Strong Research Monograph Misses Opportunity

With each of a large class of positive measures µ on the real line it is possible to associate a sequence {p n (x)} of orthogonal polynomials with the property that ∞ −∞ p m (x)p n (x)dµ(x) = 0, m = n, 1, m = n. (1) Such a sequence satisfies a three term recurrence relation xp n (x) = a n P n+1 (x) + b n P n (x) + a n−1 P n−1 (x) (2) Conversely, for suitable starting values and coefficient sequences {a n }, {b n }, the recurrence relation (2) generates a sequence of polynomials satisfying (1) for some measure µ. The polynomials have their zeros within the interval of support of the measure.