On the Range of Validity of Integral Transform Methods in Tsallis Statistical Mechanics

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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.