A finite sum representation of the Appell series F 1 ( a,b,b';c;x,y )

Abstract We use Picard's integral representation of the Appell series F 1 (a,b,b′; c; x,y) for Re (a)>0, Re (c−a)>0 to obtain a finite sum algebraic representation of F1 in the case when a, b, b′ and c are positive integers with c>a. The series converges for |x| |y| and we show that F 1 (a,b,b′; c; x,y) has two overlaying singularities at each of the points x=1 and y=1, one polar and one logarithmic in nature, when a, b, b′, c∈ N with c>a.