INTEGRAL REPRESENTATIONS FOR HORN'S H 2 FUNCTION AND OLSSON'S F P FUNCTION

We derive some Euler type double integral representations for hypergeometric functions in two variables. In the first part of this paper we deal with Horn’s H2 function, in the second part with Olsson’s FP function. Our double integral representing the FP function is compared with the formula for the same integral representing an H2 function by M. Yoshida (Hiroshima Math. J. 10 (1980), 329–335) and M. Kita (Japan. J. Math. 18 (1992), 25–74). As specified by Kita, their integral is defined by a homological approach. We present a classical double integral version of Kita’s integral, with outer integral over a Pochhammer double loop, which we can evaluate as H2 just as Kita did for his integral. Then we show that shrinking of the double loop yields a sum of two double integrals for FP.