  We find an explicit formula for the gamma vector in terms of the input
polynomial in a way that extends it to arbitrary polynomials. More
specifically, we find explicit linear combination in terms of coefficients of
the input polynomial (using Catalan numbers and binomial coefficients) and an
expression involving the derivative of the input polynomial. The first
expression suggests connections to common Coxeter group/noncrossing partition
structures in existing gamma positivity examples. In the case where the input
is the $h$-polynomial of a simplicial complex, this gives an interpretation of
the gamma vector as a measure of differences in local and global contributions.
We also apply them to connect signs/inequalities of (shifts of) the gamma
vector to upper/lower bound conditions on coefficients of the input polynomial.
Finally, we make use of the shape of the sums used to make these estimates and
connections with intersection numbers to relate these properties of the gamma
vector to algebraic structures (e.g. characteristic classes involved in
existing log concavity and Schur positivity properties).
