  The luminosity distance in the standard cosmology as given by $\Lambda$CDM
and consequently the distance modulus for supernovae can be defined by the
Pad\'e approximant. A comparison with a known analytical solution shows that
the Pad\'e approximant for the luminosity distance has an error of $4\%$ at
redshift $= 10$. A similar procedure for the Taylor expansion of the luminosity
distance gives an error of $4\%$ at redshift $=0.7 $; this means that for the
luminosity distance, the Pad\'e approximation is superior to the Taylor series.
The availability of an analytical expression for the distance modulus allows
applying the Levenberg--Marquardt method to derive the fundamental parameters
from the available compilations for supernovae. A new luminosity function for
galaxies derived from the truncated gamma probability density function models
the observed luminosity function for galaxies when the observed range in
absolute magnitude is modeled by the Pad\'e approximant. A comparison of
$\Lambda$CDM with other cosmologies is done adopting a statistical point of
view.
