Vol. 49, No. 2, 1973

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On finding the distribution function for an orthogonal polynomial set

Wm. R. Allaway

Vol. 49 (1973), No. 2, 305–310
Abstract

Let {an}n=0 and {bn}n=0 be real sequences with bn > 0 and {bn}n=0 bounded. Let {Pn(x)}n=0 be a sequence of polynomials satisfying the recurrence formula

{
xPn(x) = bn−1Pn−1(x)+ anPn (x) +bnPn+1(x) (n ≧ 0)
P  (x) = 0 P (x) = 1.
−1         0
(1.1)

Then there is a substantially unique distribution function ψ(t) with respect to which the Pn(x) are orthogonal. That is,

∫ ∞
Pn(x)Pm (x )dψ (x) = Knδn.m(n,m ≧ 0),
−∞

where Kn0 and δn,m is the kronecker delta. This paper gives a method of constructing ψ(x) for the case limn→∞b2n = 0, limn→∞b2n+1 = b < , the set of limit points of {αn}n=1 equals {−α,α} and limn→∞{a2n + a2n+1} = 0. The same method can be used in the case limn→∞bn = 0 and the set of limit points of {an}n=0 is bounded and finite in number.

Mathematical Subject Classification
Primary: 42A52
Milestones
Received: 16 August 1972
Published: 1 December 1973
Authors
Wm. R. Allaway