2 research outputs found
Root geometry of polynomial sequences I: Type
This paper is concerned with the distribution in the complex plane of the
roots of a polynomial sequence given by a recursion
, with and ,
where , , and . Our results include proof of the
distinct-real-rootedness of every such polynomial , derivation of the
best bound for the zero-set \{x\mid W_n(x)=0\ \text{for some n\ge1}\}, and
determination of three precise limit points of this zero-set. Also, we give
several applications from combinatorics and topological graph theory.Comment: 24 pages, 1 figur
Root geometry of polynomial sequences II: Type (1,0)
We consider the sequence of polynomials defined by the recursion
, with initial values and
, where are real numbers, , and . We
show that every polynomial is distinct-real-rooted, and that the roots
of the polynomial interlace the roots of the polynomial .
We find that, as , the sequence of smallest roots of the
polynomials converges decreasingly to a real number, and that the
sequence of largest roots converges increasingly to a real number. Moreover, by
using the Dirichlet approximation theorem, we prove that there is a number to
which, for every positive integer , the sequence of th smallest roots
of the polynomials converges. Similarly, there is a number to which,
for every positive integer , the sequence of th largest roots of the
polynomials converges. It turns out that these two convergence points
are independent of the numbers and , as well as . We derive explicit
expressions for these four limit points, and we determine completely when some
of these limit points coincide.Comment: 37 page