The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708. A derangement is a permutation that has no fixed points, and the derangement number is the number of fixed point free permutations on an element set. Furthermore, the derangement polynomials are natural extensions of the derangement numbers. In this paper, we study the derangement polynomials and numbers, their connections with cosine-derangement polynomials and sine-derangement polynomials, and their applications to moments of some variants of gamma random variables.

中文翻译：

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某些涉及置换多项式的恒等式和伽玛随机变量的数量和矩

排列偏差的计数问题由PierreRémondde Montmort于1708年提出。排列是没有固定点的排列，排列数是元素集上无固定点排列的数目。此外，排列多项式是排列数的自然扩展。在本文中，我们研究了排列多项式和数，它们与余弦排列多项式和正弦排列多项式的联系，以及它们在伽玛随机变量某些变体的矩中的应用。