Complete high temperature expansions for one loop finite temperature effects

We develop exact, simple closed form expressions for partition functions associated with relativistic bosons and fermions in odd spatial dimensions. These expressions, valid at high temperature, include the effects of a non-trivial Polyakov loop and generalize well-known high temperature expansions. The key technical point is the proof of a set of Bessel function identities which resum low temperature expansions into high temperature expansions. The complete expressions for these partition functions can be used to obtain one-loop finite temperature contributions to effective potentials, and thus free energies and pressures.