q-Calculus Framework For Entropy In Multifractal Distributions

The connection between Tsallis entropy for a multifractal distribution and Jackson's $q$-derivative is established. Based on this derivation and definition of a homogeneous function, a $q$-analogue of Shannon's entropy is discussed. $q$-additivity of this entropy is shown. We also define $q$-analogue of Kullback relative entropy. The implications of lattice structure beneath $q$-calculus are highlighted in the context of $q$-entropy.