Q-log-convexity from Linear Transformations and Polynomials with Only Real Zeros

Abstract In this paper, we mainly study the stability of iterated polynomials and linear transformations preserving the strong q -log-convexity of polynomials. Let [ T n , k ] n , k ≥ 0 be an array of nonnegative numbers. We give some criteria for the linear transformation y n ( q ) = ∑ k = 0 n T n , k x k ( q ) preserving the strong q -log-convexity (resp. log-convexity). As applications, we derive that some linear transformations (for instance, the Stirling transformations of two kinds, the Jacobi–Stirling transformations of two kinds, the Legendre–Stirling transformations of two kinds, the central factorial transformations, and so on) preserve the strong q -log-convexity (resp. log-convexity) in a unified manner. In particular, we confirm a conjecture of Lin and Zeng, and extend some results of Chen et al., and Zhu for strong q -log-convexity of polynomials, and some results of Liu and Wang for transformations preserving the log-convexity. The stability property of iterated polynomials implies the q -log-convexity. By applying the method of interlacing of zeros, we also present two criteria for the stability of the iterated Sturm sequences and q -log-convexity of polynomials. As consequences, we get the stabilities of iterated Eulerian polynomials of types A and B , and their q -analogs. In addition, we also prove that the generating functions of alternating runs of types A and B , the longest alternating subsequence and up–down runs of permutations form a q -log-convex sequence, respectively.

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