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

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 F-1 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 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 is an element of N with c > a, (C) 1999 Elsevier Science B.V. All rights reserved.