PDE-Systems associated with the hypergeometric functions in three variables and their particular solutions near the origin

Abstract: The great success of the theory of hypergeometric series in one variable has stimulated the development of a corresponding theory in two and more variables. Horn has investigated the convergence of 34 (14 complete and 20 confluent) hypergeometric series of two variables and established the systems of partial differential equations which they satisfy. At present, 600 (of which 205 are complete and 395 are confluent) hypergeometric functions of three second-order variables are known. The present work is devoted to the composition of systems of partial differential equations satisfied by 600 confluent hypergeometric functions of three variables. In addition, a particular solutions (if such solutions exist) of some systems of differential equations have been found near the origin.