关于下限函数和上限函数公式的一些改进
Some Refinements of Formulae Involving Floor and Ceiling Functions
摘要：
地板和天花板函数经常出现在数学中，操纵涉及地板和天花板的总和是一种微妙的游戏。幸运的是，著名的教科书《具体数学》提供了一个很好的介绍，解释了一些技术，并将一些单和或双和作为练习。对于两个这样的双和，我们提供了它们的单和类似物。这些封闭形式的身份给予的一个双重分区的多元（视为分区）的所有b进制数字的一个非负整数。我们还提出了双和单和模拟涉及分数部分函数和移位分数部分函数。
英文摘要：
The floor and ceiling functions appear often in mathematics and manipulating sums involving floors and ceilings is a subtle game. Fortunately, the well-known textbook Concrete Mathematics provides a nice introduction with a number of techniques explained and a number of single or double sums treated as exercises. For two such double sums we provide their single-sum analogues. These closed-form identities are given in terms of a dual partition of the multiset (regarded as a partition) of all b-ary digits of a nonnegative integer. We also present the double- and single-sum analogues involving the fractional part function and the shifted fractional part function.
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