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 $T_n(x)$ have only real roots, and roots of $T_n(x)$ interlace those of $T_{n+1}(x)$, as well as those of $T_{n+2}(x)$, which gives an affirmative answer to a open question of Fisk. We establish necessary and sufficient conditions such that the generalized central trinomial coefficients $T_n(b,c)$ form a log-convex sequence or a Stieltjes moment sequence.