CONNECTION FORMULAS RELATED WITH APPELL'S F2, HORN'S H2 AND OLSSON'S FP FUNCTIONS

Abstract

Some of the connection problems associated with the system of differential equations E2, which is satisfied by Appell’s F2 function, are solved by using integrals of Euler type. The present results give another proof of connection formulas related with Appell’s F2, Horn’s H2 and Olsson’s FP functions, which are obtained by Olsson. 0. Introduction Appell’s hypergeometric function F2 is the analytic continuation of F2(a, b1, b2, c1, c2; x, y)= ∑ m,n≥0 (a)m+n(b1)m(b2)n m!n!(c1)m(c2)n xm yn, |x | + |y|< 1, where (a)n = 0(a + n)/0(a), and satisfies the system E2 of rank four [AKdF, Er]: (E2)  [ x(1− x) ∂2 ∂x2 − xy ∂2 ∂x∂y + {c1 − (a + b1 + 1)x} ∂ ∂x − b1 y ∂ ∂y − ab1 ] F = 0, [ y(1− y) ∂2 ∂y2 − xy ∂2 ∂x∂y + {c2 − (a + b2 + 1)y} ∂ ∂y − b2x ∂ ∂x − ab2 ] F = 0, which is defined on the space C2\ { {x = 0} ∪ {x = 1} ∪ {y = 0} ∪ {y = 1} ∪ {x + y = 1} } ⊂ (P1)2. In [Ol], Olsson shows that a fundamental set of solutions of E2 around the point (0, 1) in the case |x/(1− y)|< 1 or that around the point (0,∞) is given by Horn’s hypergeometric function H2 and Olsson’s hypergeometric function FP , while that around the point (0, 0) is given by F2. Moreover, he also derives some connection formulas related with F2, H2 and 2010 Mathematics Subject Classification: Primary 33C60; Secondary 33C65, 33C70.