An alternative proof of a theorem of Stieltjes and related results

Let Pn(x), n ⩾ 1 be the orthogonal polynomials defined by anPn + 1(x) + an − 1Pn − 1(x) + bnPn(x) = xPn(x), P0(x) = 0, P1(x) = 1, where both sequences an and bn are bounded and an > 0. that ψ(x) is the unique (up to a constant) distribution function which corresponds to the measure of orthogonality of Pn(x) and denote by S(ψ) the spectrum of ψ(x). Alternative proofs of a theorem due to Stieltjes and of a conjecture due to Maki concerning the limit points of S(ψ) are given. A typical example to the Makiʹs conjecture together with a general result concerning the density of the zeros of the polynomials Pn(x) covers as a particular case a theorem of Chihara which generalizes the well-known theorem of Blumenthal.