Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters

We consider the asymptotic behavior of the incomplete gamma functions gamma(-a,-z) and Gamma(-a,-z) as a goes to infinity. Uniform expansions are needed to describe the transition area z~a in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform equation by applying techniques used for the existing uniform expansions for gamma(a,z) and Gamma(a,z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section

High-temperature expansion of the one-loop free energy of a scalar field on a curved background

The complete form of the high-temperature expansion of the one-loop contribution to the free energy of a scalar field on a stationary gravitational background is derived. The explicit expressions for the divergent and finite parts of the high-temperature expansion in a three-dimensional space without boundaries are obtained. These formulas generalize the known one for the stationary spacetime. In particular, we confirm that for a massless conformal scalar field the leading correction to the Planck law proportional to the temperature squared turns out to be nonzero due to non-static nature of the metric. The explicit expression for the so-called energy-time anomaly is found. The interrelation between this anomaly and the conformal (trace) anomaly is established. The natural simplest Lagrangian for the "Killing vector field" is given.Comment: 27 pages;considerable changes made,appendix B changed,equation (52) changed abstract changed, conclusion added, some references adde