Quadratic transformations of multiple hypergeometric series

Using factorization method and introducing canonical forms of multiple hypergeometric series allows all quadratic transformations for all series satisfying the corresponding applicability conditions to be obtained in the form of a small set of general basic relations. In other words any quadratic transformation for standard series, for example for the Gauss', Appell's, Horn's, Kampé de Fériet's, Lauricella's, Gelfand's series, etc., as well as for numerous nameless series, can be obtained as particular cases of the relations given in the paper. Along with completeness and generality of analysis the factorization method ensures an essential simplification of the theory by introducing a natural hierarchical structure into a system of quadratic connections between nine types of canonical forms. These results may contribute to development of advanced computer algebra systems capable to analyze, automatically, important properties of those multiple series which find large-scale applications in mathematics, mathematical physics and theoretical chemistry.