Vol. 28, No. 3, 1969

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ISSN: 0030-8730
A note on recursively defined orthogonal polynomials

Daniel Paul Maki

Vol. 28 (1969), No. 3, 611–613
Abstract

Let {ai}i=0 and {bi}i=0 be real sequences and suppose the bi’s are all positive. Define a sequence of polynomials {Pi(x)}i=0 as follows: P0(x) = 1, P1(x) = (x a0)∕b0, and for n 1

bnPn+1(x) = (x − an)Pn(x)− bn− 1Pn−1(x ).
(*)

Favard showed that the polynomials {Pi(x)} are orthonormal with respect to a bounded increasing function ψ defined on (−∞,+). This note generalizes recent constructive results which deal with connections between the two sequences {ai} and {bi} and the spectrum of ψ. (The spectrum of ψ is the set S(ψ) = {λ : ψ(λ + 𝜖) ψ(λ 𝜖) > 0 for all 𝜖 > 0}.) It is shown that if bi 0 then every limit point of the sequence {ai} is in S(ψ).

Mathematical Subject Classification
Primary: 33.40
Milestones
Received: 12 January 1968
Published: 1 March 1969
Authors
Daniel Paul Maki