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Twisted period relations for Lauricella's hypergeometric function F_A

Yoshiaki Goto

Abstract

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

Twisted period relations for Lauricella's hypergeometric function F_A

Abstract

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

Paper Structure

This paper contains 6 sections, 8 theorems, 70 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 $E_A (a,b,c)$ 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 $E_A (a,b,c)$ in a simply connected domain in $D_A -S$ 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 vectors $b^{i_1 \cdots

Theorems & Definitions (15)

  • Proposition 2.2: L, Nakayama
  • Proposition 2.3: Integral representation of Euler type, L
  • Example 4.2
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • proof
  • Remark 4.5
  • Theorem 4.6
  • proof
  • ...and 5 more