34 research outputs found
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres
On unimodality problems in Pascal's triangle
Many sequences of binomial coefficients share various unimodality properties.
In this paper we consider the unimodality problem of a sequence of binomial
coefficients located in a ray or a transversal of the Pascal triangle. Our
results give in particular an affirmative answer to a conjecture of Belbachir
et al which asserts that such a sequence of binomial coefficients must be
unimodal. We also propose two more general conjectures.Comment: 12 pages, 2 figure