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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 and study the
-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 -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 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
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