The connection between Jackson and Hausdorff derivatives in the context of generalized statistical mechanics

Abstract

In literature one can find many generalizations of the usual Leibniz derivative, such as Jackson derivative, Tsallis derivative and Hausdorff derivative. In this article we present a connection between Jackson derivative and recently proposed Hausdorff derivative. On one hand, the Hausdorff derivative has been previously associated with non-extensivity in systems presenting fractal aspects. On the other hand, the Jackson derivative has a solid mathematical basis because it is the $\overline{q}$-analog of the ordinary derivative and it also arises in quantum calculus. From a quantum deformed $\overline{q}$-algebra we obtain the Jackson derivative and then address the problem of $N$ non-interacting quantum oscillators. We perform an expansion in the quantum grand partition function from which we obtain a relationship between the parameter $\overline{q}$, related to Jackson derivative, and the parameters $\zeta$ and $q$ related to Hausdorff derivative and Tsallis derivative, respectively.