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The floor and ceiling functions appear often in mathematics and manipulatingsums involving floors and ceilings is a subtle game. Fortunately, thewell-known textbook Concrete Mathematics provides a nice introduction with anumber of techniques explained and a number of single or double sums treated asexercises. For two such double sums we provide their single-sum analogues.These closed-form identities are given in terms of a dual partition of themultiset (regarded as a partition) of all b-ary digits of a nonnegativeinteger. We also present the double- and single-sum analogues involving thefractional part function and the shifted fractional part function.

Some Refinements of Formulae Involving Floor and Ceiling Functions