  Our collective knowledge of nuclear cross sections is recorded as resonance
parameters in nuclear data libraries. To evaluate these parameters, campaigns
of measurements are fitted with a parametric model of nuclear cross sections
called R-matrix theory. In order to remove the arbitrary boundary parameters in
the Wigner-Eisenbud R-matrix parametrization, the community is considering
converting all nuclear data libraries to Brune parameters. In this article, we
show there are more Brune parameters than previously thought. Below the channel
threshold, we prove there exists two types of additional 'shadow poles' -
branch shadow poles and analytic shadow poles - depending on how we continue
R-matrix operators to complex wavenumbers (to do so we establish Mittag-Leffler
expansions of R-matrix operators). This entails there are more Brune resonance
energies than levels. Yet, we also prove that choosing any subset of Brune
poles will yield the same cross sections than using the entire set of Brune
poles, as long as it has at least the number of levels. In practice, this means
that shadow Brune poles can be discarded from the new nuclear data libraries.
Many isotopes are evaluated with the Reich-Moore approximation, introducing
complex resonance energies to eliminate certain channels. We generalize Brune's
parameterization to encompass the Reich-Moore approximation and the additional
shadow poles, and show that all Brune parameters depend on what convention we
choose to continue the R-matrix operators to complex wavenumbers. To convert
nuclear data libraries to Brune parameters, the nuclear scientists community
must thus first decide on such a convention. The authors argue in favor of
analytic continuation. The first evidence of shadow poles in Brune's
alternative parametrization of R-matrix theory is observed in isotope
xenon-134, spin-parity group 1/2(-).
