We introduce and study the cumulative information generating function, which provides a unifying mathematical tool suitable to deal with classical and fractional entropies based on the cumulative distribution function and on the survival function. Specifically, after establishing its main properties and some bounds, we show that it is a variability measure itself that extends the Gini mean semi-difference. We also provide (i) an extension of such a measure, based on distortion functions, and (ii) a weighted version based on a mixture distribution. Furthermore, we explore some connections with the reliability of $k$-out-of-$n$ systems and with stress-strength models for multi-component systems. Also, we address the problem of extending the cumulative information generating function to higher dimensions.

翻译：本文介绍并研究了累积信息生成函数，该函数提供了一个统一的数学工具，适用于处理基于累积分布函数和生存函数的经典熵与分数阶熵。具体而言，在确立其基本性质及若干界之后，我们证明该函数本身是一种变异性度量，它扩展了基尼均值半差。我们还提供了：(i) 基于扭曲函数的此类度量的推广，以及 (ii) 基于混合分布的加权版本。此外，我们探讨了该函数与 $k$-out-of-$n$ 系统可靠性以及多组件系统应力-强度模型之间的某些联系。同时，我们解决了将累积信息生成函数推广到更高维度的问题。