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Masanobu Kaneko

Publications and source records attributed to Masanobu Kaneko.

At least 19 recordsLinked to original sources
arXiv2026

The Klein-Vélu septic, 2-Division, and modular function fields of level 14

We study the 2-division of the Klein-Vélu elliptic normal septic. Its three non-zero 2-torsion points give rise to an explicit cubic equation over the function field of $X(7)$. We identify the three roots of this cubic with modular functions expressed in terms of septic theta functions at $τ/2,τ$, and $2τ$, and show that its splitting field is precisely the function field of $X(14)$. This gives an explicit link between the 2-division geometry of the Klein-Vélu septic and the passage from full level 7 to full level $14$. Several intermediate modular function fields in the resulting $S_3$-extension are also described.

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arXiv2025

On finite analogues of Euler's constant

We introduce and study finite analogues of Euler's constant in the same setting as finite multiple zeta values. We define a couple of candidate values from the perspectives of a ``regularized value of $ζ(1)$'' and of Mascheroni's and Kluyver's series expressions of Euler's constant using Gregory coefficients. Moreover, we reveal that the differences between them always lie in the $\mathbb{Q}$-vector space spanned by 1 and values of a finite analogue of logarithm at positive integers.

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arXiv2025

New Proofs of the Explicit Formulas of Arakawa--Kaneko Zeta Values and Kaneko--Tsumura $η$- and $ψ$- Values

In this paper, we establish some new identities of integrals involving multiple polylogarithm functions and their level two analogues in terms of Hurwitz-type multiple zeta (star) values. Using these identities, we provide new proofs of the explicit formulas of Arakawa--Kaneko zeta values, Kaneko--Tsumura $η$- and $ψ$-values, and also give a formula for double $T$-values.

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arXiv2024

Parametric Apéry-type Series and Hurwitz-type Multiple Zeta Values

In this paper, we extend the main results of a 2024 \emph{Advances in Applied Mathematics} paper \cite{XuZhao2021c} about Apéry-type series involving central binomial coefficients and the multiple ($t-$)harmonic sums to parametric Apéry-type series involving parametric binomial coefficients and Hurwitz-type multiple harmonic (star) sums. In particular, we will establish many explicit relations between parametric Apéry-type series involving one or two parametric binomial coefficients and Hurwitz-type multiple zeta values (with $r$-variables) by using the method of iterated integrals.

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arXiv2024

Two formulas for certain double and multiple polylogarithms in two variables

We give a weighted sum formula for the double polylogarithm in two variables, from which we can recover the classical weighted sum formulas for double zeta values, double $T$-values, and some double $L$-values. Also presented is a connection-type formula for a two-variable multiple polylogarithm, which specializes to previously known single-variable formulas. This identity can also be regarded as a generalization of the renowned five-term relation for the dilogarithm.

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arXiv2023

Elliptic normal curves of even degree and theta functions

An elliptic curve may be immersed in ${\mathbf{P}}^{N-1}$ as a degree $N$ curve using level $N$ structure. In the case where $N$ is odd, there are well known classical results dating back to Bianchi and Klein. In this paper we study the case of even $N$ in some detail. In particular, over the complex number field, we define an immersion using suitably chosen theta functions, and study the quadratic equations satisfied by them.

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arXiv2022

A study of a Fuchsian system of rank 8 in 3 variables and the ordinary differential equations as its restrictions

A Fuchsian system of rank 8 in 3 variables with 4 parameters is presented. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential equation of order four with three singular points. A middle convolution of this equation turns out to be the tensor product of two Gauss hypergeometric equation, and another middle convolution sends this equation to the Dotsenko-Fateev equation. Local solutions to these ordinary differential equations are found. Their coefficients are sums of products of the Gamma functions. These sums can be expressed as special values of the generalized hypergeometric series $_4F_3$ at 1. Keywords: Fuchsian differential equation, hypergeometric differential equation, middle convolution, Dotsenko-Fateev equation, recurrence formula, series solution

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arXiv2021

On finite multiple zeta values of level two

We introduce and study a ``level two'' analogue of finite multiple zeta values. We give conjectural bases of the space of finite Euler sums as well as that of usual finite multiple zeta values in terms of these newly defined elements. A kind of ``parity result'' and certain sum formulas are also presented.

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arXiv2020

A generalized regularization theorem and Kawashima's relation for multiple zeta values

Kawashima's relation is conjecturally one of the largest classes of relations among multiple zeta values. Gaku Kawashima introduced and studied a certain Newton series, which we call the Kawashima function, and deduced his relation by establishing several properties of this function. We present a new approach to the Kawashima function without using Newton series. We first establish a generalization of the theory of regularizations of divergent multiple zeta values to Hurwitz type multiple zeta values, and then relate it to the Kawashima function. Via this connection, we can prove a key property of the Kawashima function to obtain Kawashima's relation.

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arXiv2019

On poly-cosecant numbers

We introduce and study a `level two' generalization of the poly-Bernoulli numbers, which may also be regarded as a generalization of the cosecant numbers. We prove a recurrence relation, two exact formulas, and a duality relation for negative upper-index numbers.

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arXiv2019

Quasi-derivation relations for multiple zeta values revisited

We take another look at the so-called quasi-derivation relations in the theory of multiple zeta values, by giving a certain formula for the quasi-derivation operator. In doing so, we are not only able to prove the quasi-derivation relations in a simpler manner but also give an analog of the quasi-derivation relations for finite multiple zeta values.

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arXiv2019

On a variant of multiple zeta values of level two

We study a variant of multiple zeta values of level 2, which forms a subspace of the space of alternating multiple zeta values. This variant, which is regarded as the `shuffle counterpart' of Hoffman's `odd variant', exhibits nice properties such as duality, shuffle product, parity results, etc., like ordinary multiple zeta values. We also give some conjectures on relations between our values, Hoffman's values, and multiple zeta values.

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arXiv2018

Zeta functions connecting multiple zeta values and poly-Bernoulli numbers

We first review our previous works of Arakawa and the authors on two, closely related single-variable zeta functions. Their special values at positive and negative integer arguments are respectively multiple zeta values and poly-Bernoulli numbers. We then introduce, as a generalization of Sasaki's work, level 2 analogue of one of the two zeta functions and prove results analogous to those by Arakawa and the first named author.

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arXiv2016

A new integral-series identity of multiple zeta values and regularizations

We present a new "integral=series" type identity of multiple zeta values, and show that this is equivalent in a suitable sense to the fundamental theorem of regularization. We conjecture that this identity is enough to describe all linear relations of multiple zeta values over Q. We also establish the regularization theorem for multiple zeta-star values, which too is equivalent to our new identity. A connection to Kawashima's relation is discussed as well.

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arXiv2016

Multi-poly-Bernoulli numbers and related zeta functions

We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at non-positive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the $ξ$-function defined by Arakawa and the first-named author. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.

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