  We report measurements of the specific heat of
Ba$_{0.59}$K$_{0.41}$Fe$_{2}$As$_{2}$, an Fe-pnictide superconductor with $T_c$
= 36.9 K, for which there are suggestions of an unusual electron pairing
mechanism. We use a new method of analysis of the data to derive the parameters
characteristic of the electron contribution. It is based on comparisons of
${\alpha}$-model expressions for the electron contribution with the total
measured specific heat, which give the electron contribution directly. It
obviates the need in the conventional analyses for an independent, necessarily
approximate, determination of the lattice contribution, which is subtracted
from the total specific heat to obtain the electron contribution. It eliminates
the uncertainties and errors in the electron contribution that follow from the
approximations in the determination of the lattice contribution. Our values of
the parameters characteristic of the electron contribution differ significantly
from those obtained in conventional analyses of specific-heat data for five
similar hole-doped BaFe$_{2}$As$_{2}$ superconductors, which also differ
significantly among themselves. They show that the electron density of states
is comprised of contributions from two electron bands with
superconducting-state energy gaps that differ by a factor 3.8, with 77$\%$
coming from the band with the larger gap. The variation of the specific heat
with magnetic field is consistent with extended $s$-wave pairing, one of the
theoretical predictions. The relation between the densities of states and the
energy gaps in the two bands is not consistent with a theoretical model based
on interband interactions alone. Comparison of the normal-state density of
states with band-structure calculations shows an extraordinarily large
effective mass enhancement, for which there is no precedent in similar
materials and no theoretical explanation.
