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.
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