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

✍ Scribed by A. Cuyt; K. Driver; J. Tan; B. Verdonk

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 82 KB
• Volume: 105
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(99)00035-7

✦ Synopsis

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 ÿnite 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.