  An identity of Chung, Graham and Knuth involving binomial coefficients and
Eulerian numbers motivates our study of a class of polynomials that we call
binomial-Eulerian polynomials. These polynomials share several properties with
the Eulerian polynomials. For one thing, they are $h$-polynomials of simplicial
polytopes, which gives a geometric interpretation of the fact that they are
palindromic and unimodal. A formula of Foata and Sch\"utzenberger shows that
the Eulerian polynomials have a stronger property, namely $\gamma$-positivity,
and a formula of Postnikov, Reiner and Williams does the same for the
binomial-Eulerian polynomials. We obtain $q$-analogs of both the
Foata-Sch\"utzenberger formula and an alternative to the
Postnikov-Reiner-Williams formula, and we show that these $q$-analogs are
specializations of analogous symmetric function identities. Algebro-geometric
interpretations of these symmetric function analogs are presented.
