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In this paper we investigate analytic properties of sextet polynomials of hexagonal systems. For the pyrene chains, we show that zeros of the sextet polynomials $P_n(x)$ are real, located in the open interval $(-3-2\sqrt{2},-3+2\sqrt{2})$ and dense in the corresponding closed interval. We also show that coefficients of $P_n(x)$ are symmetric, unimodal, log-concave, and asymptotically normal. For general hexagonal systems, we show that real zeros of all sextet polynomials are dense in the interval $(-\infty,0]$, and conjecture that every sextet polynomial has log-concave coefficients., Comment: 18 pages, 6 figures

We present a characterization of eigenvalue inequalities between two Hermitian matrices by means of inertia indices. As applications, we deal with some classical eigenvalue inequalities for Hermitian matrices, including the Cauchy interlacing theorem and the Weyl inequality, in a simple and unified approach. We also give a common generalization of eigenvalue inequalities for (Hermitian) normalized Laplacian matrices of simple (signed, weighted, directed) graphs. Our approach is also suitable for Hermitian matrices of the second kind of digraphs recently introduced by Mohar., Comment: to appear in Linear and Multilinear Algebra

By means of the derivative operator and Chu-Vandermonde convolution, four families of summation formulas involving harmonic numbers with even or odd indexes are established.

By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored. They give some new hypergeometric series identities when the parameters are specified.

In this paper we prove the strong $q$-log-convexity of the Eulerian polynomials of Coxeter groups using their exponential generating functions. Our proof is based on the theory of exponential Riordan arraya and a criterion for determining the strong $q$-log-convexity of polynomials sequences, whose generating functions can be given by the continued fraction. As consequences, we get the strong $q$-log-convexity the Eulerian polynomials of type $A_n,B_n$, their $q$-analogous and the generalized Eulerian polynomials associated to the arithmetic progression $\{a,a+d,a+2d,a+3d,\ldots\}$ in a unified manner., Comment: 13pages