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 Jc(n,k;z)Jc(n,k;z) and JS(n,k;z)JS(n,k;z) is a Pólya frequency sequence if and only if z∈[−1,1]z∈[−1,1] and study the z-total positivity properties of these numbers. Moreover, the polynomial sequences{∑k=0nJS(n,k;z)yk}n⩾0and{∑k=0nJc(n,k;z)yk}n⩾0 are proved to be strongly {z,y}{z,y}-log-convex. In the same vein, we extend a recent result of Chen et al. about the Ramanujan polynomials to Chapotonʼs 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.