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We study the structures of Pfaffian equations and contiguity relations of the hypergeometric function of type $(k+1,k+n+2)$ by using twisted cohomology groups and the intersection form on them. We apply our results to algebraic statistics; numerical evaluation of the normalizing constants of two way contingency tables with fixed marginal sums.

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Pfaffian Equations and Contiguity Relations of the Hypergeometric Function of Type (&lt;i&gt;k&lt;/i&gt;+1, &lt;i&gt;k&lt;/i&gt;+&lt;i&gt;n&lt;/i&gt;+2) and Their Applications