Article ID Journal Published Year Pages File Type
6417895 Journal of Mathematical Analysis and Applications 2014 26 Pages PDF
Abstract

This paper obtains several evaluations of multivariate hypergeometric functions for particular parameter values and at special algebraic points. They have a high interest not only on their own, but also in the light of the remarkable implications for both pure mathematics and mathematical physics. Following our research started in [30] and [31], we provide some contribution to such functions' computability inside and outside their disks of convergences. In the first part we provide some new results in the spirit of Theorem 3.1 of [31], obtaining formulae for the values of multivariate hypergeometric functions by generalizing a well known identity of Kummer [23], to the hypergeometric functions of two or more variable like those of Appell and Lauricella denoted FD(n). In the second part, using some reduction schemes of hyperelliptic integrals due to Goursat [16], Hermite [18,19] we evaluate Appell and Lauricella's FD(n) hypergeometric functions and their analytic continuations at some particular locations. Finally, by exploiting reductions of hyperelliptic integrals to elliptic due to Belokolos et al. [5], Eilbeck and Enol'skii [11], Enol'skii and Kostov [12] and by Maier [27], we obtain further links from multivariate hypergeometric functions, to complete elliptic integrals and to π. We thus reach a conceptual settlement of the piece of research started by us in [30] and [31].

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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