Some Refinements of Formulae Involving Floor and Ceiling Functions

TL;DR

This paper refines and extends formulas involving floor and ceiling functions, providing new identities and analogues for sums with these functions, including those involving fractional parts, based on digit partitions.

Contribution

It introduces novel closed-form identities and analogues for sums involving floor, ceiling, and fractional part functions, expanding the mathematical toolkit for such sums.

Findings

New single-sum analogues for double sums involving floor and ceiling functions

Closed-form identities based on digit partitions of integers

Analogues involving fractional and shifted fractional parts

Abstract

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.