  Hirschman and Widder introduced a class of P\'olya frequency functions given
by linear combinations of one-sided exponential functions. The members of this
class are probability densities, and the class is closed under convolution but
not under pointwise multiplication. We show that, generically, a polynomial
function of such a density is a P\'olya frequency function only if the
polynomial is a homothety, and also identify a subclass for which each
positive-integer power is a P\'olya frequency function. We further demonstrate
connections between the Maclaurin coefficients, the moments of these densities,
and the recovery of the density from finitely many moments, via Schur
polynomials.
