A Counterexample to an Assertion Due to Blumenthal

Let the orthogonal polynomials Pn(x) be defined by Pn(x) = (x− cn)Pn−1(x)− λnPn−2(x), P−1(x) = 0, P0(x) = 1, where λn+1 > 0, cn real, limn→∞ cn = c and limn→∞ λn = λ, (n ≥ 1). Blumenthal has proved that the true interval of orthogonality [σ, τ ] of the above polynomials is given by σ = c − 2 √ λ, τ = c + 2 √ λ and the zeros of Pn(x) are dense in [σ, τ ]. Blumenthal also asserted that the spectrum of the distribution function ψ corresponding to the polynomials Pn has at most finite points in the complement of [σ, τ ]. In other words the limit points of the zeros of the polynomials Pn outside the intrval [σ, τ ] are finite. The falseness of this assertion has been proved first in 1968 with the use of a series of results concerning chain sequences and a theorem due to Szegö. Now although one can find many other ways of proving this in the literature, a concrete counter example fails to exist. Here a counterexample is given which proves the invalidity of Blumenthal’s assertion. This example is also of some interest because it exhibits a particular class of the Pollaczek polynomials where the support of the measure of orthogonality is extented beyond the interval [−1, 1].