  For a Riemann surface of genus $g\ge 2$ there exists a unique Green's
function $G_{N}(x,y)$ which transforms as a weight $N\ge 2$ form in $x$ and a
weight $1-N$ form in $y$ and is meromorphic in $x$, with a unique simple pole
at $x=y$, but is not meromorphic in $y$. For a Schottky uniformized Riemann
surface we consider meromorphic extensions of $G_{N}(x,y)$ called Green's
Functions with Extended Meromorphicity or GEM forms. GEM forms are meromorphic
in both $x$ and $y$ with a unique simple pole at $x=y$, transform as weight
$N\ge 2$ forms in $x$ but as weight $1-N$ quasiperiodic forms in $y$. We give a
reformulation of the bijective Bers map and describe a choice of GEM form with
an associated canonical basis of normalized holomorphic $N$-forms. We describe
an explicit differential operator constructed from $N=2$ GEM forms giving the
variation with respect to moduli space parameters of a punctured Riemann
surface. We also describe a new expression for the inverse Bers map.
