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Generating functions and generalized Dedekind sums. (English) Zbl 1036.11504

Electron. J. Comb. 4, No. 2, Research paper R11, 17 p. (1997); printed version J. Comb. 4, No. 2, 137-153 (1997).
Summary: We study sums of the form \(\sum_\zeta R(\zeta)\), where \(R\) is a rational function and the sum is over all \(n\)th roots of unity \(\zeta\) (often with \(\zeta=1\) excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for \(\prod_\zeta(1-xR(\zeta))\). Multisection can be used to evaluate some simple, but important sums. Finally, the method of partial fractions reduces the evaluation of arbitrary generalized Dedekind sums to those of a very simple form.

MSC:

11F20 Dedekind eta function, Dedekind sums
05A15 Exact enumeration problems, generating functions
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