On the range of validity of Hilhorst-type formulae

Abstract

We show that if we require positive definite probabilities, then frequently cited results of Prato and Hilhorst on the nonextensive equilibrium statistical mechanics of gases is valid only for a limited range of the Tsallis parameter, q. We determine the range of validity of the Hilhorst and the Prato formulae. We then use various integral representations of the Gamma function to derive a new formula whose range of validity is wider than that of the Hilhorst and the Prato formulae. We then apply the new formula to the classical ideal gas.