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In reaction rate theory, in production-destruction type models and inreaction-diffusion problems when the total derivatives are replaced byfractional derivatives the solutions are obtained in terms of Mittag-Lefflerfunctions and their generalizations. When fractional calculus enters into thepicture the solutions of these problems, usually available in terms ofgeneralized hypergeometric functions, switch to Mittag-Leffler functions andtheir generalizations into Wright functions and subsequently into Foxfunctions. In this paper, connections are established among generalizedMittag-Leffler functions, Mathai's pathway model, Tsallis statistics,Beck-Cohen superstatistics, and among corresponding entropic measures. TheMittag-Leffler function, for large values of the parameter, approaches apower-law. For values of the parameter close to zero, the Mittag-Lefflerfunction behaves like a stretched exponential. The Mittag-Leffler function is ageneralization of the exponential function and represents a deviation from theexponential paradigm whenever it shows up in solution of physical problems. Thepaper elucidates the relation between analytic representations of theq-exponential function that is fundamental to Tsallis statistics,Mittag-Leffler, Wright, and Fox functions, respectively, utilizingMellin-Barnes integral representations.

Functions of Mittag-Leffler and Fox: The Pathway Model to Tsallis Statistics and Beck-Cohen Superstatistics