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Quadratic relations for confluent hypergeometric functions. (English) Zbl 1006.33004
Let $ \omega$ be a $ 1 $-form on $ \Bbb{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=\Bbb{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.
33C15Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
Full Text: DOI
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