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