Integral representations for Horn's $H_2$ function and Olsson's $F_P$ function

TL;DR

This paper derives Euler-type double integral representations for Horn's $H_2$ and Olsson's $F_P$ hypergeometric functions, comparing and connecting different integral formulations.

Contribution

It introduces a classical double integral form for Kita's homological integral of Olsson's $F_P$ function, linking it with Horn's $H_2$ function through loop shrinking.

Findings

Derived Euler-type double integrals for $H_2$ and $F_P$ functions.

Connected Kita's homological integral with a classical double integral.

Showed how shrinking the loop yields a sum of two $F_P$ integrals.

Abstract

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 $H_2$ function, in the second part with Olsson's $F_P$ function. Our double integral representing the $F_P$ function is compared with the formula for the same integral representing an $H_2$ 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 $H_2$ just as Kita did for his integral. Then we show that shrinking of the double loop yields a sum of two double integrals for $F_P$.