Some Refinements of Formulae Involving Floor and Ceiling Functions

Luka Podrug; Dragutin Svrtan

Vol. 28 No. 2 (2023)

Regular Articles

Some Refinements of Formulae Involving Floor and Ceiling Functions

Published 2023-10-19

Luka Podrug
Dragutin Svrtan

Luka Podrug
University of Zagreb, Faculty of Civil Engineering

Dragutin Svrtan
University of Zagreb, Faculty of Science

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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.

Keywords

sum, floor function, ceiling function, fractional part function

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