  We present a systematic short time expansion for the generating function of
the one point height probability distribution for the KPZ equation with droplet
initial condition, which goes much beyond previous studies. The expansion is
checked against a numerical evaluation of the known exact Fredholm determinant
expression. We also obtain the next order term for the Brownian initial
condition. Although initially devised for short time, a resummation of the
series allows to obtain also the \textit{long time large deviation function},
found to agree with previous works using completely different techniques.
Unexpected similarities with stationary large deviations of TASEP with periodic
and open boundaries are discussed. Two additional applications are given. (i)
Our method is generalized to study the linear statistics of the {Airy point
process}, i.e. of the GUE edge eigenvalues. We obtain the generating function
of the cumulants of the empirical measure to a high order. The second cumulant
is found to match the result in the bulk obtained from the Gaussian free field
by Borodin and Ferrari, but we obtain systematic corrections to the Gaussian
free field (higher cumulants, expansion towards the edge). This also extends a
result of Basor and Widom to a much higher order. We obtain {large deviation
functions} for the {Airy point process} for a variety of linear statistics test
functions. (ii) We obtain results for the \textit{counting statistics of
trapped fermions} at the edge of the Fermi gas in both the high and the low
temperature limits.
