Ullman and Erdös
and Freud have studied the distribution of the zeros of certain classes of
orthogonal polynomials. Among other results they have shown that for a wide
class of weight functions the associated orthogonal polynomials all have the
same limiting zero distribution. We show, in a related result, that in certain
cases one can deduce the limiting distribution of the zeros of the orthogonal
polynomials without explicitly knowing the weight function (or the distribution
function) of the orthogonal polynomials. In particular, for polynomials with
certain types of triple recurrence formula we show that the limiting zero
distribution is always the one studied by Ullman and Erdös and Freud.
Polynomials with this limiting zero distribution are said to have “regular zero
behavior”.
In §2 we give the basic definitions and notation needed for our result.
Our main theorem is in §3, and §4 contains some related comments and
examples.
