Pastro, Pier Ivan Cohomological interpretation of some classic formulas satisfied by hypergeometric functions. (English) Zbl 0786.33005 Funkc. Ekvacioj, Ser. Int. 36, No. 1, 11-43 (1993). Because of Euler’s formula \[ _ 2F_ 1 \left[ {{a,\;b} \atop c};z\right]= \text{const.} \int_ \sigma x^{b-1} (1-x)^{c-b-1} (1- zx)^{-a} dx \] (where \(\sigma\) is the Pochhammer loop of integration) the hypergeometric function \({}_ 2F_ 1\) can be considered as a period of a differential form of a family of variables depending on \(z\) i.e. the Riemann surface of the integrand function. In the article under review the cohomology of such families are treated.Some formulas are studied: the quadratic formula \[ _ 2 F_ 1 \left[ {{a/2,\;(a+1)/2} \atop {b+1/2}}; (z/(z-2))^ 2\right]= (1-z/2)^ a\;{}_ 2F_ 1 \left[ {{a,\;b} \atop {2b}}; z\right] \] and the “reduction” formula \[ _ 3F_ 2 \left[ {{a,\;b,\;f} \atop {f\;c}};z\right] ={}_ 2F_ 1 \left[ {{a,\;b} \atop c}; z\right] \] where \({}_ 3F_ 2\) is the generalized hypergeometric function. These formulas are proved by means of morphisms between the families of varieties involved because of Euler’s integral representation. Reviewer: P.I.Pastro (Ponzano) MSC: 33C20 Generalized hypergeometric series, \({}_pF_q\) Keywords:morphisms between families of variables PDFBibTeX XMLCite \textit{P. I. Pastro}, Funkc. Ekvacioj, Ser. Int. 36, No. 1, 11--43 (1993; Zbl 0786.33005)