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A note on \(p\)-adic \(q\)-Dedekind sums. (English) Zbl 0982.11064

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 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.

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11F20 Dedekind eta function, Dedekind sums
11B68 Bernoulli and Euler numbers and polynomials

Citations:

Zbl 0039.03801
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