Respuesta:
Let P,(x), n ~> 1 be the orthogonal polynomials defined by
anPn+l(X) -~ an_lPn_l(x) + b.P,(x) = xP,(x), Po(x) = 0, Pl(x) = 1,
where both sequences a, and b, are bounded and a. > 0.
Assume that ~,(x) is the unique (up to a constant) distribution function which corresponds to the measure of
orthogonality of P,(x) and denote by S(~,) the spectrum of ~p(x). Alternative proofs of a theorem due to Stieltjes and of
a conjecture due to Maki concerning the limit points of S(~) are given. A typical example to the Maki's conjecture
together with a general result concerning the density of the zeros of the polynomials P,(x) covers as a particular case
a theorem of Chihara which generalizes the well-known theorem of Blumenthal.
Keywords: Orthogonal polynomials; Measure of orthogonality; Zeros; Limit points of the spectrum
AMS Classification: 42C05
1. Introduction
In [7] Chihara has formulated and proved the following theorem which follows from a result of
Stieltjes [16], concerning continued fractions.
Theorem 1.1 (Stieltjes [16]). Let R. be the set of orthogonal polynomials defined by
a,R,+l(x) + a,-1R,-l(x) = xR,(x), n >~ 1,
(1.1)
Ro(x) = O, RI(x ) = 1.
Then a necessary and sufficient condition for the associated distribution function t~ to have a denumer
