Analytic aspects of generalized central trinomial coefficients

Abstract

The divisibility and congruence of usual and generalized central trinomial coefficients have been extensively investigated. The present paper is devoted to analytic properties of these numbers. We show that usual central trinomial polynomials Tn(x) have only real roots, and roots of Tn(x) interlace those of Tn+1(x), as well as those of Tn+2(x), which gives an affirmative answer to an open question of Fisk. We establish necessary and sufficient conditions such that the generalized central trinomial coefficients Tn(b,c) form a log-convex sequence or a Stieltjes moment sequence.