Extension of Chu–Vandermonde Identity and Quadratic Transformation Conditions

In 1812, Gauss stated the following identity: F12(a,b;c;1)=Γ(c)Γ(c−a−b)Γ(c−a)Γ(c−b), where, in the real case, c−a−b>0 and as an immediat consequence the Chu–Vandermonde identity: F12(a,−n;c;1)=(c−a)n(c)n for any positive integer n. In this paper, we investigate the case when c−a−b<0 by taking c=2b=−2n, n and a are positive integers (c−a−b=−n−a<0). We give two significant applications stemming from these findings. The second part of the paper will be devoted to Kummer’s conditions concerning hypergeometric quadratic transformations, particularly focusing on the distinctions between the conditions provided by Gradshteyn and Ryzhik (GR) and those by Erdélyi, Magnus, Oberhettinger, and Tricomi (EMOI) are outlined. We establish that the conditions given by GR differ from those of EMOI, and we explore the methodologies employed by both groups in deriving their results. This leads us to conclude that the search for exact and unified conditions remains an open problem.