Identities for sums of floor and ceiling functions of rational numbers

Research into sums involving floor or ceiling functions has been a popular focus in number theory.This paper investigates formulas for sums of the type n k=1 F pk q , where F represents either the floor or ceiling function, and p, q are coprime integers.Generalizing from Palatsang et al's work in 2021, we could derive explicit simple formulas, verify their uniqueness, and identify certain conditions of p and q under which these formulas are invalid.