Human and automated approaches for finite trigonometric sums

The resolvent kernel on the discrete circle and twisted cosecant sums

“…Several of those sums seem not to have known evaluations, but it is possible to establish reciprocal relations, see for example [BC13], [Ch18] or [MS20]. There also seems to be a growing interest in the evaluation of trigonometric sums in the combinatorics literature, in particular illustrated by the recent contributions [dFGK18], [EL21], [GLY22], [AZ23], and [CHJSV23] relevant for our paper. Spectral graph theory and discrete Green functions have been important topics for quite some time, see for example [Ba79] and [CY00].…”

“…In [AH18] the starting point is to use varioius results in the theory of certain special functions. Also, a discrete form of sampling theorem was used in [Ha08], while [AZ23] describes an "automated approach" for proving some trigonometric identities.…”