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Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials. (English) Zbl 1281.05011

Summary: Generalizing recent results of E. S. Egge [Eur. J. Comb. 31, No. 7, 1735–1750 (2010; Zbl 1213.05005)] and P. Mongelli [Adv. Appl. Math. 48, No. 2, 354–364 (2012; Zbl 1237.05007)], we show that each diagonal sequence of the Jacobi-Stirling numbers \(Jc(n,k;z\)) and \(JS(n,k;z\)) is a Pólya frequency sequence if and only if \(z\in [-1,1]\) and study the \(z\)-total positivity properties of these numbers. Moreover, the polynomial sequences
\[ \biggl \{\sum_{k=0}^{n}JS(n,k;z)y^k\biggr\}_{n\geq 0}\;\text{and}\;\biggl \{\sum_{k=0}^{n}Jc(n,k;z)y^k\biggr\}_{n\geq 0} \]
are proved to be strongly \(\{z,y\}\)-log-convex. In the same vein, we extend a recent result of W. Y. C. Chen et al. [Can. Math. Bull. 54, No. 2, 217–229 (2011; Zbl 1239.05190)] 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 G. A. Kalugin and D. J. Jeffrey [C. R. Math. Acad. Sci., Soc. R. Can. 33, No. 2, 50–56 (2011; Zbl 1266.11052)].

MSC:

05A15 Exact enumeration problems, generating functions
05A05 Permutations, words, matrices
05A20 Combinatorial inequalities
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
11B73 Bell and Stirling numbers
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References:

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