Trigonometric sums through Ramanujan’s theory of theta functions

Human and automated approaches for finite trigonometric sums

“…Here we describe or adapt these results. We also confirm two conjectures given by the authors of [32] at the end of their paper.…”

“…The authors of [32] cite a similar derivation in [5,Corollary 2.4]. It is interesting to note that the "disguised" equality above is already given (actually only for n even) under a slightly different disguise by Riesz in 1914 in [46,47] where it is used to prove a theorem of Bernstein (about Riesz' results also see [60, Volume II, p. 10, Eq.…”

“…At the end of their paper [32], the authors give two conjectures about which they wrote: We feel that these two can also be tackled with Ramanujan's theory although we were unable to do so. First we recall these conjectures.…”

“…Such trigonometric sums occur in a large variety of domains, and are addressed with a large variety of methods: compiling all the pertinent papers in the domain would lead at least to a book (look for example at the long list of references given in [5]). Various methods have been used, some of them in papers cited above: interpolation formulas [46,47,2,33] (to which we can add, e.g., [10,1]...), Ramanujan's theta function [7,32,51], calculus of residues (see, among several other papers, the papers by Cvijović or Cvijović et al cited in [14]; also see [30]), expansions in partial fractions (see, e.g., [13,53,12]), discrete Fourier analysis (see [3]), etc.…”

“…(1.31) of Corollary 7 in [38], and Eq. (1.11) in [32]): Much more modestly we stumbled upon the recent paper [32] published in the Ramanujan journal, where the authors give an interesting way to obtain a closed form for several trigonometric sums by using Ramanujan's theory of theta functions. In particular Theorem 1.1 in that paper gives six formulas about which the authors write that these six identities seem to be new.…”