We introduce a fractional generalization of Tsallis entropy by acting with a $q$-Caputo operator on the generating family $\sum_i p_i^{\,x}$ evaluated at $x=1$. Concretely, we define $S_{q}^α$ through the $q$-Caputo differintegral of order $0<α<1$ and derive a closed series representation in terms of the $q$-Gamma function. The construction is anchored at the evaluation point, which ensures well-behaved limits: as $α\!\to\!1$ we recover the standard Tsallis entropy $S_q$. Finally we perform a numerical calculation to show the regions where the obtained $q$-fractional entropy $S^α_q$ can be non-negative (or negative) through the fractional parameter $α$ and the non extensive index $q$.

翻译：我们通过将$q$-Caputo算子作用于生成族$\sum_i p_i^{\,x}$在$x=1$处的取值，引入了一种Tsallis熵的分数阶推广。具体而言，我们定义了通过阶数为$0<α<1$的$q$-Caputo分数阶微分积分算子得到的$S_{q}^α$，并推导出其在$q$-Gamma函数框架下的闭式级数表示。该构造以评估点为锚点，确保了良好的极限性质：当$α\!\to\!1$时，我们恢复标准的Tsallis熵$S_q$。最后通过数值计算展示了所得$q$-分数阶熵$S^α_q$通过分数参数$α$和非广延指数$q$可能呈现非负（或负）值的区域。