Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy
Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy
The q-sum x⊕qy≡x+y+(1−q)xy (x⊕1y=x+y) and the q-product x⊗qy≡[x1−q+y1−q−1]1∕(1−q) (x⊗1y=xy) emerge naturally within nonextensive statistical mechanics. We show here how they lead to two-parameter (namely, q and q′) generalizations of the logarithmic and exponential functions (noted, respectively, lnq,q′x and eq,q′x), as well as of the Boltzmann-Gibbs-Shannon entropy SBGS≡−k∑i=1Wpilnpi (noted Sq,q′). The …