New decomposition formulas associated with the Lauricella multivariable hypergeometric functions

Decomposition formulas associated with the Lauricella multivariable hypergeometric functions were known, however, due to the recurrence of those formulas, additional difficulties may arise in the applications. Further study of the properties of the famous expansion formulas showed that it can be reduced to a more convenient form. In addition, this paper contains applications of new expansion formulas to the solving of boundary value problems for a multidimensional elliptic equation with several singular coefficients.