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Quadratic relations for confluent hypergeometric functions. (English) Zbl 1006.33004

Let ω be a 1-form on 1 with poles x={x 1 ,,x m } of order n 1 ,,n m (n=n 1 ++n m ), and let ±ω =d±ω be the integrable connections on X= 1 x. Consider the twisted cohomology groups H 1 (Ω (x), ±ω ) for the complexes ±ω :Ω (x)Ω (x), where Ω k (x) is the vector space of rational k-forms with poles at most at x and consider u(t)=cexp( t ω) which satisfies -ω u(t)=0. An integral φ,γ= γ u(t)φ for some φH 1 (Ω (x), ω ) is called the hypergeometric integral, where γ is an element of twisted homology group H 1 (C ω , ω ), which is proved to be dual to H 1 (Ω (x), ω ) by the pairing ,. The authors introduce the cohomological intersection pairing for (n-2)-dimensional vector spaces H 1 (Ω (x), ω ) and H 1 (Ω (x), -ω ), and the homological intersection pairing for H 1 (C ω , ω ) and H 1 (C -ω , -ω )· For a choice of bases {φ μ ± } μ and {γ μ ± } μ of the groups H 1 (Ω (x), ±ω ) and H 1 (C ±ω , ±ω ), define the four matrices of size n-2:

Π + =φ μ + ,γ ν + μ,ν ,Π - =φ μ - ,γ ν - μ,ν ,I ch =φ μ + ,φ ν - μ,ν ,I h =γ μ + ,γ ν - μ,ν ·

The main result of this paper is the following.

Theorem. We have twisted period relations:

Π + t I h -1 t Π - =I ch

which give quadratic relations among confluent hypergeometric integrals.

MSC:
33C15Confluent hypergeometric functions, Whittaker functions, 1 F 1