On computing some special values of multivariate hypergeometric functions

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].