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，该函数为处理基于累积分布函数和生存函数的经典及分数阶熵提供了统一的数学工具。具体而言，在建立其主要性质与若干界之后，我们证明其本身是一种推广了基尼均值半差异的变异性度量。我们还提供了：（i）基于失真函数的该度量的推广形式，以及（ii）基于混合分布的加权版本。此外，我们探讨了其与$k$-out-of-$n$系统可靠性及多组件系统应力-强度模型之间的关联，并研究了将累积信息生成函数推广至高维空间的问题。