A $q$-Caputo Fractional Generalization of Tsallis Entropy

In this work we build a $q$-Caputo fractional generalization of the Tsallis entropy functional $S_q$. The construction starts from the already known Jackson derivative representation of $S_q$ and replaces the Jackson derivative with a Caputo type $q$-fractional operator ${}^C{D}_q^{\alpha }$. The parameter $q$ keeps its standard role as the nonextensive index, while the fractional parameter $\alpha$ gives a nonlocal, memory like contribution in the entropy generating procedure. The resulting functional, denoted by $S_q^{\alpha }$, is written as a convergent series of logarithms in the probabilities and is defined for $0<\alpha <1$, $q>0$, and $p_i>0$. We show explicitly that the standard Tsallis entropy is recovered in the local limit $\alpha \to 1$. We then start rewriting $S_q^{\alpha }$ in trace form and analyze its main structural properties. Non-negativity is tested numerically in a binary and ternary probability distributions. These tests show positive values in the respective domains, but they are interpreted just as numerical evidence rather than as a global/formal proof. Concavity is studied through the second derivative of the trace form contribution yielding to a parameter dependent condition. A global statement requires control over the full probability domain. We also discuss Lesche stability and expansibility. Both properties depend on the behavior of the fully resumed functional near its boundary $p\to 0^{+}$, because the entropy contains powers of $\ln p_i$. Finally, we derive the composition rule for statistically independent subsystems. The logarithmic structure produces mixed moments, so the standard Tsallis pseudo-additive law is not directly inherited for $0<\alpha <1$. As an illustration, we evaluate the proposed entropy in a two-state fractional relaxation process with memory.