  We consider a number of questions regarding the Luttinger-Ward functional and
the many-body perturbation series expansion of the proper self-energy
$\Sigma(\mathbf{k};z)$ specific to uniform ground states (ensemble of states)
of interacting fermion systems in terms of skeleton self-energy diagrams and
the interacting Green function $G(\mathbf{k};z)$. Utilising a link between the
latter series expansion and the classical moment problem (of the Hamburger
type), along with the associated continued-fraction expansion, we reaffirm our
earlier observation (2007) that for lattice models of fermions interacting
through short-range two-body potentials (i.e. for Hubbard-like models) this
series is uniformly convergent for almost all wave vectors $\mathbf{k}$ and
complex energies $z$. The limit of this series is unique. We inquire into the
reasons underlying the contrary observation by Kozik et al. (2015) regarding
skeleton-diagrammatic perturbation series expansion of $\Sigma$. In doing so,
we make a number of observations of general interest. Chief amongst these, we
observe that contrary to general belief thermal correlation functions
calculated in the energy domain (i.e. over the set of the relevant Matsubara
frequencies) are generally unreliable, in contrast to those calculated in the
imaginary-time domain. In an appendix we present a symbolic computational
formalism and short programs for singling out topologically distinct proper
self-energy diagrams that are algebraically equivalent up to determinate
multiplicative constants. This complements the work presented in our previous
publication (2019). [Abridged abstract]
