Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials

Generalizing recent results of Egge and Mongelli, we show that each diagonal sequence of the Jacobi-Stirling numbers \js(n,k;z) and \JS(n,k;z) is a P\'olya frequency sequence if and only if $z\in [-1, 1]$ and study the $z$-total positivity properties of these numbers. Moreover, the polynomial sequences \biggl\{\sum_{k=0}^n\JS(n,k;z)y^k\biggr\}_{n\geq 0}\quad \text{and} \quad \biggl\{\sum_{k=0}^n\js(n,k;z)y^k\biggr\}_{n\geq 0} are proved to be strongly $\{z,y\}$-log-convex. In the same vein, we extend a recent result of Chen et al. about the Ramanujan polynomials to Chapoton's generalized Ramanujan polynomials. Finally, bridging the Ramanujan polynomials and a sequence arising from the Lambert $W$ function, we obtain a neat proof of the unimodality of the latter sequence, which was proved previously by Kalugin and Jeffrey.Comment: 17 pages, 2 tables, the proof of Lemma 3.3 is corrected, final version to appear in Advances in Applied Mathematic

The qq-log-convexity of Domb's polynomials

In this paper, we prove the $q$-log-convexity of Domb's polynomials, which was conjectured by Sun in the study of Ramanujan-Sato type series for powers of $\pi$. As a result, we obtain the log-convexity of Domb's numbers. Our proof is based on the $q$-log-convexity of Narayana polynomials of type $B$ and a criterion for determining $q$-log-convexity of self-reciprocal polynomials.Comment: arXiv admin note: substantial text overlap with arXiv:1308.273

On the qq-log-convexity conjecture of Sun

In his study of Ramanujan-Sato type series for $1/\pi$, Sun introduced a sequence of polynomials $S_n(q)$ as given by $$S_n(q)=\sum\limits_{k=0}^n{n\choose k}{2k\choose k}{2(n-k)\choose n-k}q^k,$$ and he conjectured that the polynomials $S_n(q)$ are $q$-log-convex. By imitating a result of Liu and Wang on generating new $q$-log-convex sequences of polynomials from old ones, we obtain a sufficient condition for determining the $q$-log-convexity of self-reciprocal polynomials. Based on this criterion, we then give an affirmative answer to Sun's conjecture

Radon Numbers for Trees

Many interesting problems are obtained by attempting to generalize classical results on convexity in Euclidean spaces to other convexity spaces, in particular to convexity spaces on graphs. In this paper we consider $P_3$-convexity on graphs. A set $U$ of vertices in a graph $G$ is $P_3$-convex if every vertex not in $U$ has at most one neighbour in $U$. More specifically, we consider Radon numbers for $P_3$-convexity in trees. Tverberg's theorem states that every set of $(k-1)(d+1)-1$ points in $\mathbb{R}^d$ can be partitioned into $k$ sets with intersecting convex hulls. As a special case of Eckhoff's conjecture, we show that a similar result holds for $P_3$-convexity in trees. A set $U$ of vertices in a graph $G$ is called free, if no vertex of $G$ has more than one neighbour in $U$. We prove an inequality relating the Radon number for $P_3$-convexity in trees with the size of a maximal free set.Comment: 17 pages, 13 figure

A note on Reeb dynamics on the tight 3-sphere

We show that a nondegenerate tight contact form on the 3-sphere has exactly two simple closed Reeb orbits if and only if the differential in linearized contact homology vanishes. Moreover, in this case the Floquet multipliers and Conley-Zehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4-space.Comment: 20 pages, no figure