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Twisted cycles and twisted period relations for Lauricella's hypergeometric function F_C

Yoshiaki Goto

Abstract

We study Lauricella's hypergeometric function F_C by using twisted (co)homology groups. We construct twisted cycles with respect to an Euler-type integral representation of F_C. These cycles correspond to 2^m linearly independent solutions to the system of differential equations annihilating F_C. Using intersection forms of twisted (co)homology groups, we obtain twisted period relations which give quadratic relations for Lauricella's F_C.

Twisted cycles and twisted period relations for Lauricella's hypergeometric function F_C

Abstract

We study Lauricella's hypergeometric function F_C by using twisted (co)homology groups. We construct twisted cycles with respect to an Euler-type integral representation of F_C. These cycles correspond to 2^m linearly independent solutions to the system of differential equations annihilating F_C. Using intersection forms of twisted (co)homology groups, we obtain twisted period relations which give quadratic relations for Lauricella's F_C.

Paper Structure

This paper contains 6 sections, 8 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Differential equations and integral representations
  3. Twisted homology groups
  4. Twisted cycles corresponding to local solutions $f_{i_1 \cdots i_r}$
  5. Intersection numbers of twisted cohomology groups
  6. Twisted period relations

Key Result

Proposition 2.2

The system $E_{C} (a,b,c)$ is a holonomic system of rank $2^m$ with the singular locus If $c_1 ,\ldots ,c_m \not\in \mathbb{Z}$, then the vector space of solutions to $E_{C} (a,b,c)$ in a simply connected domain in $D_C -S$ is spanned by the following $2^m$ elements: Here $r$ runs from $0$ to $m$, indices $i_{1} ,\ldots ,i_{r}$ satisfy $1\leq i_{1} <\cdots <i_{r} \leq m$, and the row vector $c^{

Theorems & Definitions (18)

  • Proposition 2.2: HT, L
  • Proposition 2.3: Integral representation of Euler type, Example 3.1 in AK
  • Example 4.2
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • proof
  • Remark 4.5
  • Theorem 4.6
  • proof
  • ...and 8 more