On symmetric polynomials with only real zeros and nonnegative γ-vectors

Abstract

We present a family of polynomials with nonnegative coefficients that are symmetric and have only real nonpositive zeros. Therefore, they are also log-concave, unimodal and have nonnegative γ-vectors. This family of polynomials arises by applying the operator (ZrDs)n to the rational function 11−z. Two distinguished examples are the classical Eulerian polynomials and the Narayana polynomials. We use two special linear bases, besides the standard basis, to provide various characterizations for these polynomials and their coefficients, in terms of generalized Stirling numbers of the second kind. The type of generalized Stirling numbers we deal with, emerge from the normal ordering of (ZrDs)n. We generalize several formulas known for Eulerian and Narayana polynomials. Likewise, we provide an explicit formula to compute the so-called γ-vector of any symmetric polynomial of the type consider in this paper. Further related polynomials are obtained by applying the operator (ZrDs)n to a general Möbius transformation. Along the way, associated families of novel nonnegative integer sequences are put in evidence as a byproduct of our polynomial characterizations. We provide explicit formulas for some of these integral sequences.