Composition law of $\kappa$-entropy for statistically independent systems

The intriguing and still open question concerning the composition law of $\kappa$-entropy $S_{\kappa }\left(f\right)=\frac{1}{2\kappa }{\sum }_i(f_i^{1-\kappa }-f_i^{1+\kappa })$ with $0<\kappa <1$ and ${\sum }_i{f}_i=1$ is here reconsidered and solved. It is shown that, for a statistical system described by the probability distribution $f=\left\{f_{ij}\right\}$, made up of two statistically independent subsystems, described through the probability distributions $p=\left\{p_i\right\}$ and $q=\left\{q_j\right\}$, respectively, with $f_{ij}=p_i{q}_j$, the joint entropy $S_{\kappa }\left(pq\right)$ can be obtained starting from the $S_{\kappa }\left(p\right)$ and $S_{\kappa }\left(q\right)$ entropies, and additionally from the entropic functionals $S_{\kappa }(p/e_{\kappa })$ and $S_{\kappa }(q/e_{\kappa }),e_{\kappa }$ being the $\kappa$-Napier number. The composition law of the $\kappa$-entropy is given in closed form and emerges as a one-parameter generalization of the ordinary additivity law of Boltzmann-Shannon entropy recovered in the $\kappa \to 0$ limit.