Transformation of the Triple Series of Gelfand, Graev, and Retakh into a Series of the Same Type and Related Problems

A transformation of the triple series $T$ related to the Grassmanian $G_{2,4}$ into a series of the same structure type is obtained. This transformation generalizes the reduction formula of Gelfand, Graev, and Retakh taking the series $T$ to the Gauss function under two additional conditions and two more general reduction formulas taking the series $T$ to the Appell function $F_1$ and to the Horn function $G_2$ under one of the additional conditions. The approach used to analyze the series $T$ is based on the representation of the initial series $T$ in terms of series with convenient computational properties.