arXiv subjects

Jyoichi Kaneko

Publications and source records attributed to Jyoichi Kaneko.

6 recordsLinked to original sources

arXiv2025

The monodromy representation of a hypergeometric system in $m$ variables of rank $p^m$

We study the monodromy representation of the hypergeometric system $\mathcal{F}_{C}^{p,m}(a,B)$ in $m$ variables of rank $p^m$ with parameters $a$ and $B$. This system can be regarded as a multi-variable model of the generalized hypergeometric equation of rank $p$. We construct $m+1$ loops which generate the fundamental group of the complement of the singular locus of $\mathcal{F}_{C}^{p,m}(a,B)$, and we show that they satisfy certain relations as elements of the fundamental group. We produce circuit matrices along these loops with respect to a fundamental system of solutions to $\mathcal{F}_C^{p,m}(a,B)$ under certain non-integrality conditions on parameters $a$ and $B$.

math.CA↗

arXiv2024

A system of hypergeometric differential equations in $m$ variables of rank $p^m$

We define a hypergeometric series in $m$ variables with $p+(p-1)m$ parameters, which reduces to the generalized hypergeometric series $_pF_{p-1}$ when $m=1$, and to Lauricella's hypergeometric series $F_C$ in $m$ variables when $p=2$. We give a system of hypergeometric differential equations annihilating the series. Under some non-integral conditions on parameters, we give an Euler type integral representation of the series, and linearly independent $p^m$ solutions to this system around a point near to the origin. We show that this system is of rank $p^m$, and determine its singular locus.

math.CA↗

arXiv2022

$q$-Selberg Integrals and Koornwinder Polynomials

We prove a generalization of the $q$-Selberg integral evaluation formula. The integrand is that of $q$-Selberg integral multiplied by a factor of the same form with respect to part of the variables. The proof relies on the quadratic norm formula of Koornwinder polynomials. We also derive generalizations of Mehta's integral formula as limit cases of our integral.

math.CA↗

arXiv2017

The fundamental group of the complement of the singular locus of Lauricella's $F_C$

We study the fundamental group of the complement of the singular locus of Lauricella's hypergeometric function $F_C$ of $n$ variables. The singular locus consists of $n$ hyperplanes and a hypersurface of degree $2^{n-1}$ in the complex $n$-space. We derive some relations that holds for general $n\geq 3$. We give an explicit presentation of the fundamental groupin the three-dimensional case. We also consider a presentation of the fundamental group of $2^3$-covering of this space. In the version 2, we omit some of the calculations. For all the calculations, refer to the version 1 (arXiv:1710.09594v1) of this article.

math.AG↗

arXiv2016

A system of hypergeometric differential equations in two variables of rank 9

We study a hypergeometric function in two variables and a system of hypergeometric differential equations associated with this function. This is a regular holonomic system of rank $9$. We give a fundamental system of solutions to this system in terms of this hypergeometric series. We give circuit matrices along generators of the fundamental group of the complement of its singular locus with respect to our fundamental system.

math.AG↗

arXiv2015

Pfaffian of Appell's hypergeometric system $F_4$ in terms of the intersection form of twisted cohomology groups

We study a Pfaffian of the system of differential equations annihilating Appell's hypergeometric series $F_4(a,b,c;x)$ by twisted cohomology groups associated with integrals representing solutions to this system. We simplify its connection matrix by the pull-back under a double cover of the complement of the singular locus. We express the simplified connection matrix in terms of the intersection form of the twisted cohomology groups.

math.AG↗