Linear Transformations and Reduction Formulas for the Gelfand Hypergeometric Functions Associated with the Grassmannians \(G_{2,4} \) and \(G_{3,6} \)

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We show that the Gelfand hypergeometric functions associated with the Grassmannians \(G_{2,4} \) and \(G_{3,6} \) with some special relations imposed on the parameters can be represented in terms of hypergeometric series of a simpler form. In particular, a function associated with the Grassmannian \(G_{2,4} \) (the case of three variables) can be represented (depending on the form of the additional conditions on the parameters of the series) in terms of the Horn series \(H_2 ,G_2 \), of the Appell functions \(F_1 ,F_2 ,F_3 \) and of the Gauss functions \(F_1^2 \), while the functions associated with the Grassmannian \(G_{3,6} \) (the case of four variables) can be represented in terms of the series \(G_2 ,F_1 ,F_2 ,F_3 \) and \(F_1^2 \). The relation between certain formulas and the Gelfand--Graev--Retakh reduction formula is discussed. Combined linear transformations and universal elementary reduction rules underlying the method were implemented by a computer program developed by the authors on the basis of the computer algebra system Maple V-4.