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Cohomological interpretation of some classic formulas satisfied by hypergeometric functions. (English) Zbl 0786.33005

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.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
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