Composition law of k-entropy for statistically independent systems

The intriguing and still open problem regarding the composition law of $\kappa$-entropy $S_{\kappa}(f)=\frac{1}{2\kappa}\sum_i (f_i^{1-\kappa}-f_i^{1+\kappa})$ with $0<\kappa<1$ and $\sum_i f_i =1$ is reconsidered and solved. It is shown that for a statistical system described by the probability distribution $f=\{ f_{ij}\}$, composed by two statistically independent subsystems described through the probability distributions $p=\{ p_i\}$ and $q=\{ q_j\}$ respectively with $f_{ij}=p_iq_j$, the joint entropy $S_{\kappa}(p\,q)$ can be obtained starting from the entropies $S_{\kappa}(p)$ and $S_{\kappa}(q)$ 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 a closed form and emerges as one-parameter generalization of the ordinary additivity law of Boltzmann-Shannon entropy recovered in the $\kappa \rightarrow 0$ limit.