On computing some special values of multivariate hypergeometric functions (Q401111)

The authors establish interesting formulas for the values of the multivariate hypergeometric functions, including Appell and Lauricella \(F^{(n)}_D\) functions, at some special points inside and outside their disks of convergence. Relationships of hypergeometric functions' special values to certain complete elliptic integrals are also discussed. The basic tools are a well-established generalization of Kummer identity to multivariate hypergeometric functions, the usual representation for such functions and some reduction schemes of hyperelliptic integrals.