A class of relativistic dispersion functions for unmagnetized thermal plasmas is defined by generalizing functions first defined by Trubnikov in 1958. Recursion relations are derived that allow one to generate explicit expressions for the class of functions in terms of the relativistic plasma dispersion function T(z, ρ) introduced by Godfrey et al. in 1975. These functions are relevant to the description of the response of a weakly mangetized, highly relativistic, thermal plasma.