  We reconsider the Gram-Hadamard bound as it is used in constructive quantum
field theory and many body physics to prove convergence of Fermionic
perturbative expansions. Our approach uses a recursion for the amplitudes of
the expansion, discovered originally by Djokic arXiv:1312.1185. It explains the
standard way to bound the expansion from a new point of view, and for some of
the amplitudes provides new bounds, which avoid the use of Fourier transform,
and are therefore superior to the standard bounds for models like the cold
interacting Fermi gas.
