Gessel, Ira M. 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. Cited in 17 Documents MSC: 11F20 Dedekind eta function, Dedekind sums 05A15 Exact enumeration problems, generating functions Keywords:generalized Dedekind sums; factorization; multisection; partial fractions PDFBibTeX XMLCite \textit{I. M. Gessel}, Electron. J. Comb. 4, No. 2, Research paper R11, 17 p. (1997); printed version J. Comb. 4, No. 2, 137--153 (1997; Zbl 1036.11504) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: Array a(n,m) = ((n+2)/2)^m*Sum_{k=1..n+1} 1/sin(k*Pi/(n+2))^(2m), n>=0, k>=0, read by ascending antidiagonals.