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Genki Shibukawa

Publications and source records attributed to Genki Shibukawa.

At least 19 recordsLinked to original sources
arXiv2026

Notes on twisted homology and cohomology groups for the Wirtinger integral

The Wirtinger integral is one of the integral representations of the Gauss hypergeometric function. Its integrand is given by a product of complex powers of theta functions. We study the structure of the twisted homology and cohomology groups associated with this integral. Using the involution on the complex torus, we show that these groups decompose into eigenspaces which are orthogonal with respect to the intersection forms. Each eigenspace is related to the twisted (co)homology group associated with the Euler-type integral representation of the Gauss hypergeometric function. We also show that the corresponding intersection matrices admit simple forms.

math.AG↗
arXiv2025

The intertwining property for $β$-Laguerre processes and integral operators for Jack polynomials

The aim of this paper is to study intertwining relations for Laguerre process with inverse temperature $β\ge 1$ and parameter $α>-1$. We introduce a Markov kernel that depends on both $β$ and $ α$, and establish new intertwining relations for the $β$-Laguerre processes using this kernel. A key observation is that Jack symmetric polynomials are eigenfunctions of our Markov kernel, which allows us to apply a method established by Ramanan and Shkolnikov. Additionally, as a by-product, we derive an integral formula for multivariate Laguerre polynomials and multivariate hypergeometric functions associated with Jack polynomials.

math.PR↗
arXiv2024

The generalized Zwegers' $μ$-function and transformation formulas for the bilateral basic hypergeometric series

By applying Slater's transformation formulas for the bilateral basic hypergeometric series ${}_2ψ_{2}$, we derive three type translation formulas for the generalized Zwegers' $μ$-function (``continuous $q$-Hermite function'') which was introduced by Shibukawa--Tsuchimi (SIGMA, 2023). From some Bailey's transformation formula of ${}_2ψ_{2}$, we also give a formula for the expression of the generalized Zwegers' $μ$-function by some very-well-poised bilateral basic hypergeometric series ${}_4ψ_{8}$. As an application of this new expression formula for the generalized Zwegers' $μ$-function, we obtain some new Fourier expansions for the Weierstrass and Jacobi elliptic functions.

math.CA↗
arXiv2023

Curious congruences for cyclotomic polynomials II

We promote the recent research by Akiyama and Kaneko on the higher-order derivative values $Φ_n^{(k)}(1)$ of the cyclotomic polynomials. This article focuses on Lehmer's explicit formula of $Φ_n^{(k)}(1)/Φ_n(1)$ as a polynomial of the Euler and Jordan totient functions over $\mathbb{Q}$. Then we prove Akiyama-Kaneko's conjecture that the polynomials have a specific simple factor.

math.NT↗
arXiv2023

A Generalization of Zwegers' $μ$-Function According to the $q$-Hermite-Weber Difference Equation

We introduce a one parameter deformation of the Zwegers' $μ$-function as the image of $q$-Borel and $q$-Laplace transformations of a fundamental solution for the $q$-Hermite-Weber equation. We further give some formulas for our generalized $μ$-function, for example, forward and backward shift, translation, symmetry, a difference equation for the new parameter, and bilateral $q$-hypergeometric expressions. From one point of view, the continuous $q$-Hermite polynomials are some special cases of our $μ$-function, and the Zwegers' $μ$-function is regarded as a continuous $q$-Hermite polynomial of ''$-1$ degree''.

math.CA↗
arXiv2022

Principal Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups

In this paper, we study the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at \[ \zeta_{(n,k)} := ( 1, \zeta_n, \zeta_n^2, \dots, \zeta_n^{kn-1} ), \] where $\zeta_n$ is a primitive $n$th root of unity. We give explicit formulas for several classes of special values. We also show that these special values naturally appear as the coefficients in the expansion of the $k$th power of the circulant determinant of order $n$ (the group determinant of the cyclic group of order $n$). These results extend Ore's formulas for the case $k = 1$. Furthermore, we determine the number of terms in the $k$th power of the group permanent of the cyclic group of order $n$. This extends Brualdi and Newman's result for $k = 1$.

math.RT↗
arXiv2021

An explicit formula of powers of the $2\times 2$ quantum matrices and its applications

We present an explicit formula of the powers for the $2\times 2$ quantum matrices, that is a natural quantum analogue of the powers of the usual $2\times 2$ matrices. As applications, we give some non-commutative relations of the entries of the powers for the $2\times 2$ quantum matrices, which is a simple proof of the results of Vokos-Zumino-Wess (1990).

math.RA↗
arXiv2020

Raising type twisted Pieri formulas for Jack polynomials and their applications to interpolation Jack polynomials

We propose new Pieri type formulas for Jack polynomials, which is another kind of Pieri type formulas than the ones in the previous paper (G. Shibukawa, arXiv:2004.12875). From these new Pieri type formulas, we give yet another proof of difference and Pieri formulas for interpolation Jack polynomials. Further, we also generalize a falling type twisted Pieri formula to the binomial type polynomials including Jack and multivariate Bernoulli polynomials.

math.CO↗
arXiv2020

A revisit to periodic continuants

We give a simple proof of some explicit formulas of periodic continuants by Chebyshev polynomials of the second kind given by Rozsa.

math.AP↗
arXiv2019

Multivariate Bernoulli polynomials

We introduce a multivariate analogue of Bernoulli polynomials and give their fundamental properties: difference and differential relations, symmetry, explicit formula, inversion formula, multiplication theorem, and binomial type formula. Further, we consider a multivariate analogue of the multiple Bernoulli polynomials and give their fundamental properties.

math.CA↗
arXiv2019

Some arithmetic properties of an elliptic Dedekind sum

We give an explicit expression of the elliptic classical Dedekind sum which is a special case of multiple elliptic Dedekind sums introduced by Egami. We also determine the denominator of the rational part and zeros of the elliptic classical Dedekind sum.

math.NT↗