Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy Available to Purchase

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 remarkable properties of the $(q,q′)$-generalized logarithmic function make the entropic form $Sq,q′≡k∑i=1Wpilnq,q′(1∕pi)$ satisfy, for large regions of $(q,q′)$⁠, important properties such as expansibility, concavity, and Lesche stability, but not necessarily composability.