Reciprocity formulae for multiple Dedekind-Rademacher sums
Let \(n\geq 2\), \(1\leq k\leq n\), \(a_1,\ldots,a_n,r_1,\ldots,r_k,\ldots,r_n\in{\mathbb N}\), \(\overrightarrow{A_k}=(a_1,\ldots,\check{a}_k,\ldots,a_n), \overrightarrow{R_k}=(r_1,\ldots,\check{r}_k,\ldots,r_n)\in{\mathbb N}^{n-1}\). The multiple Dedekind-Rademacher sums are defined by \[ S_n(\overrightarrow{A_k}, \overrightarrow{R_k})=\sum_{t=0}^{a_k-1} \prod_{1\leq j\neq k\leq n}\bar{B}_{r_j}\left(\frac{a_jt}{a_k}\right), \] where \(\bar{B}_n(z)\) are Bernoulli's functions. Let \[ {\mathfrak S}_n(\overrightarrow{A_k},\overrightarrow{\Phi_k}):= \sum_{\overrightarrow{R_k}\in{\mathbb N}^{n-1} }\frac{S_n(\overrightarrow{A_k}, \overrightarrow{R_k})}{r_1!\ldots \check{r}_k!\ldots r_n!}\prod_{1\leq j\neq k\leq n}\left(\frac{2\pi i\varphi_j}{a_j}\right)^{r_j-1}, \] where \(\overrightarrow{\Phi_k}=(\varphi_1,\ldots,\check{\varphi}_k,\ldots,\varphi_n)\in{\mathbb R}^{n-1}\). The authors prove: If \(a_1,\ldots,a_n\) are pairwise coprime, \(\varphi_1,\ldots,\varphi_n\in(-1,1), \varphi_1+\ldots+\varphi_n=0\), then \[ \sum_{k=1}^n {\mathfrak S}_n(\overrightarrow{A_k},\overrightarrow{\Phi_k})=-\frac{1}{(2i)^{n-1}}\frac{\sin(\pi\sum_{j=1}^n\frac{\varphi_j}{a_j})}{\prod_{j=1}^n\sin(\pi\frac{\varphi_j}{a_j})} \]
scientific article; zbMATH DE number 5860044
scientific article