A note on \(p\)-adic \(q\)-Dedekind sums (Q2762962)

Let \(p\) be an odd prime number, \({\mathbb{Q}} _p\) the field of \(p\)-adic numbers, \({\mathbb{Z}} _p\) the ring of \(p\)-adic integers, and \({\mathbb{C}} _p\) the completion of an algebraic closure \(\bar {\mathbb{Q}} _p\) of \({\mathbb{Q}} _p\) with respect to the prolongation, say \(v _p\), of the normalized \(p\)-adic absolute value of \({\mathbb{Q}} _p\). Also, let \(q\) be an element of \({\mathbb{C}} _p\) satisfying the inequality \(|q - 1|_p < 1\), where \(|\cdot |_p\) denotes the absolute value of \({\mathbb{C}} _p\) continuously extending \(v _p\). The paper under review gives a \(p\)-adic \(q\)-analog to the higher-order Dedekind sum \(S _m (h, k)\) introduced by \textit{T. M. Apostol} (for each triple of positive integers \(m, h\) and \(k\)) [see Duke Math. J. 17, 147-157 (1950; Zbl 0039.03801)] in the special case where \(k\) is relatively prime to the product \(p\cdot h\). This is obtained by constructing a certain continuous \(p\)-adic \(q\)-function with prescribed values on the set of positive integers.