A theorem of Gil', relating the zeros of entire functions of finite order to traces of powers of matrices, is generalized to entire functions of finite rank and then analyzed from the point of view of spectral theory. Plenty of relevant examples are given, including a generalization of Viete's relations for the elementary symmetric functions of the roots of a polynomial.
"Identities for the zeros of entire functions of finite rank and spectral theory." Rocky Mountain J. Math. 49 (4) 1049 - 1062, 2019. https://doi.org/10.1216/RMJ-2019-49-4-1049