  We present new analytic continuation formulas for the Appell
$F_4(a,b;c,d;x,y)$ double hypergeometric series where $d=a-b+1$, which allows
quadratic transformations of the Gauss ${}_2F_1$ hypergeometric function to be
used in the intermediate steps of the derivation. Such formulas are of
relevance to loop calculations of quantum field theory where they can been
used, for instance, to obtain new series representations of the two-loop
massive sunset Feynman diagram. The analytic continuation procedure introduced
in this paper is also sufficiently general so as to find uses elsewhere.
