  We consider elliptic partial differential equations with diffusion
coefficients that depend affinely on countably many parameters. We study the
summability properties of polynomial expansions of the function mapping
parameter values to solutions of the PDE, considering both Taylor and Legendre
series. Our results considerably improve on previously known estimates of this
type, in particular taking into account structural features of the affine
parametrization of the coefficient. Moreover, the results carry over to more
general Jacobi polynomial expansions. We demonstrate that the new bounds are
sharp in certain model cases and we illustrate them by numerical experiments.
