A note on recursively defined orthogonal polynomials

Abstract

Let {J-o and {t}= 0 ^e rea l sequences and suppose the bi, are all positive. Define a sequence of polynomials {P;(sc)}=o as follows: P 0 (cc) = 1, Pi(x) = (x -ao)lb o , and for n ^ 1 (*) bnPn+i(x) = (a -)iV;) -&-iP-i(O . Favard showed that the polynomials {P (x)} are orthonormal with respect to a bounded increasing function defined on (-co, +oo). This note generalizes recent constructive results which deal with connections between the two sequences {a } and {bi} and the spectrum of f. (The spectrum of is the set S() = { : f( + e)-f( -e)>0 for all > 0}.) It is shown that if bi -> 0 then every limit point of the sequence {^} is in S(f).