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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.
##### MSC:
 33C15 Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
Full Text:
##### References:
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