A finite sum representation of the Appell series F1(a,b,b′;c;x,y)

摘要

We use Picard's integral representation of the Appell series F1(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|<1,|y|<1 and we show that F1(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.