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

Let \( \omega\) be a \( 1 \)-form on \( \mathbb{P}^1\) with poles \(x=\{ x_1, \dots, x_m \}\) of order \( n_1, \dots, n_m\) \((n = n_1 + \cdots + n_m)\), and let \( \nabla_{\pm\omega} = d \pm \omega \wedge \) be the integrable connections on \( X=\mathbb{P}^1 \setminus x\). Consider the twisted cohomology groups \( H^1(\Omega^\bullet(x),\nabla_{\pm\omega}) \) for the complexes \(\nabla_{\pm\omega}:\Omega^\bullet(x) \to \Omega^\bullet(x) \), where \( \Omega^k(x) \) is the vector space of rational \( k\)-forms with poles at most at \(x\) and consider \( u(t) = c \exp (\int^t \omega)\) which satisfies \(\nabla_{ - \omega}u(t)=0 \). An integral \( \langle \varphi, \gamma \rangle= \int_\gamma u(t)\varphi \) for some \( \varphi \in H^1(\Omega^\bullet(x),\nabla_\omega) \) is called the hypergeometric integral, where \( \gamma \) is an element of twisted homology group \( H_1(C_\bullet^{\omega}, \partial_\omega)\), which is proved to be dual to \(H^1(\Omega^\bullet(x),\nabla_\omega) \) by the pairing \( \langle\;, \;\rangle \). The authors introduce the cohomological intersection pairing for \( (n-2)\)-dimensional vector spaces \(H^1(\Omega^\bullet(x),\nabla_{\omega}) \) and \( H^1(\Omega^\bullet(x),\nabla_{-\omega}) \), and the homological intersection pairing for \( H_1(C_\bullet^{\omega}, \partial_{\omega}) \) and \( H_1(C_\bullet^{-\omega}, \partial_{-\omega}) .\) For a choice of bases \(\{ \varphi^\pm_\mu\}_\mu\) and \(\{ \gamma^\pm_\mu\}_\mu\) of the groups \(H^1(\Omega^\bullet(x),\nabla_{\pm\omega}) \) and \( H_1(C_\bullet^{\pm\omega},\partial_{\pm \omega}) \), define the four matrices of size \( n-2:\) \[ \Pi^+ = \langle \varphi^+_\mu, \gamma^+_\nu \rangle_{\mu, \nu}, \quad \Pi^- = \langle \varphi^-_\mu, \gamma^-_\nu \rangle_{\mu, \nu}, \quad I_{\text{ch}}= \langle \varphi^+_\mu, \varphi^-_\nu \rangle_{\mu, \nu}, \quad I_{\text{h}}= \langle \gamma^+_\mu, \gamma^-_\nu \rangle_{\mu, \nu}. \] The main result of this paper is the following.
Theorem. We have twisted period relations: \[ \Pi^+ {}^t I_{\text{h}}^{-1} {}^t\Pi^- = I_{\text{ch}} \] which give quadratic relations among confluent hypergeometric integrals.

MSC:

33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
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