A construction of q-analogue of Dedekind sums

Abstract

If one looks back the classical proof (cf. Carlitz [4]) of the reciprocity law for Dedekind sums in order to construct q-analogue of Dedekind sums which also have the reciprocity law, one can soon see that the following elementary equation is essential in the proof:for any distinct complex numbers u and v, where means the generating function of Euler numbers associated to u. So we must extend the above equation to the generating function of q-Euler numbers for our purpose.