Some properties of Horn type second order double hypergeometric series

Horn [1931, Hypergeometrische Funktionen zweier Veranderlichen, Math. Ann.,105(1), 381-407], (corrections in Borngasser [1933, Uber hypergeometrische funkionen zweier Veranderlichen, Dissertation, Darmstadt], defined and investigated ten second order hypergeometric series of two variables). In the course of further investigation of Horn’s series, we noticed the existence of hypergeometric double series $H^\ast_2$ analogous to Horn’s double series  $H^\ast_2$. The principal object of this paper is to present a natural further step toward the mathematical properties and presentations concerning the analogous hypergeometric double series $H^\ast_2$. Indeed, motivated by the important role of the Horn’s functions in several diverse fields of physics and the contributions toward the unification and generalization of the hyper-geometric functions, we establish a system of partial differential equations, integral representations, expansions, analytic continuation, transformation formulas and generating relations. Also, we discuss the links for the various results, which are presented in this paper, with known results.