A note on generalized Dedekind sums (Q2652086)

The author generalizes Rademacher's identity (see the review Zbl 0057.03801 above) by putting \(f\left(\frac rk\right)=\frac rk-\left[\frac rk\right]-\frac 12+\frac 1{2k}\) and defining \[ S_n(h_1,h_2,\ldots,h_n;k)=\sum_{r_1,\ldots, r_n\pmod k} f\left(\frac {r_1}{k}\right)\cdots \left(\frac {r_n}{k}\right) f\left(\frac{r_1h_1+\ldots +r_nh_n}{k}\right). \] Then \(n+2\) of the above \(S_n\)'s satisfy an identity from which Rademacher's follows by putting \(n=1\).