Analytical continuation formulas for multiple hypergeometric series

Applying canonical forms of multiple hypergeometric series along with the use of the operator factorization method makes it possible to obtain, in explicit and most general form, the analytical continuation formulas directly applicable to arbitrary series having the Gaussian type ($2/\!/1$) with respect to one or several arguments. The formulas help us to unify a great number of particular formulas scattered throughout the literature. Moreover they give us a complete set of relations for any non-standard series provided that it pertains to the Gaussian type with respect to at least one of its variables. Due to simplicity and universality of the basic relations there arises an important possibility to implement computer-aided analysis of numerous repeated transformations with respect to different arguments of the series and to join these transformations with other important types of transformations. This possibility may have significance for mathematical analysis, mathematical physics, computer algebra and theoretical chemistry.