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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 allowsquadratic transformations of the Gauss ${}_2F_1$ hypergeometric function to beused in the intermediate steps of the derivation. Such formulas are ofrelevance to loop calculations of quantum field theory where they can beenused, for instance, to obtain new series representations of the two-loopmassive sunset Feynman diagram. The analytic continuation procedure introducedin this paper is also sufficiently general so as to find uses elsewhere.

New analytic continuations for the Appell $F_4$ series from quadratic transformations of the Gauss $_{2}F_1$ function