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

V. Schwammle, C. Tsallis

The $q$-sum $x{\oplus }_{q}y\equiv x+y+(1-q)xy$ ($x{\oplus }_{1}y=x+y$) and the $q$-product $x{\otimes }_{q}y\equiv [x^{1-q}+y^{1-q}-1{]}^{\frac{1}{1-q}}$ ($x{\otimes }_{1}y=xy$) emerge natur…