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Integral representations for Horn’s \(H_2\) function and Olsson’s \(F_P\) function. (English) Zbl 1447.33009

Authors’ 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, 329–335 (1980; Zbl 0441.33011)] and M. Kita [Jpn. J. Math., New Ser. 18, No. 1, 25–74 (1992; Zbl 0767.33009)]. 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 \(F_P\).

MSC:

33C65 Appell, Horn and Lauricella functions
33C05 Classical hypergeometric functions, \({}_2F_1\)
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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