On reciprocity formula of Apostol-Dedekind sum with quasi-periodic Euler functions

•We obtain the reciprocity formula for the Apostol-Dedekind sum with quasi-periodic Euler functions.•The Boole summation formula is used in our proof.•A concrete example for the reciprocity formula has been presented.

The Apostol-Dedekind sum with quasi-periodic Euler functions is an analogue of Apostol's definition of the generalized Dedekind sum with periodic Bernoulli functions. In this paper, using the Boole summation formula, we shall obtain the reciprocity formula for this sum.