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Yoshihiro Takeyama

Publications and source records attributed to Yoshihiro Takeyama.

At least 19 recordsLinked to original sources
arXiv2026

Applications of a formula of Maesaka-Seki-Watanabe type for multiple harmonic $q$-sums

Maesaka, Seki and Watanabe proved a formula for multiple harmonic sums. Yamamoto generalized it to Schur-type multiple harmonic sums, and the second author proved a $q$-analogue of this generalization. In this paper, we give two applications of the $q$-analogue formula. The first is an alternative proof of the duality of a $q$-analogue of multiple zeta values. The second is a proof of an identity for a $q$-analogue of the Kawashima function.

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arXiv2025

An identity for generating series of deformations of multiple zeta values within an algebraic framework

Bachmann proves an identity expressing the generating series of MacMahon's generalized sum-of-divisors $q$-series in terms of Eisenstein series. MacMahon's $q$-series can be regarded as a $q$-analogue of the multiple zeta value $ζ(2, 2, \ldots , 2)$, up to a power of $1-q$. Based on this observation, we generalize Bachmann's identity within an algebraic framework and prove a general identity. As a byproduct, we obtain a formula for the generating series of another deformation of multiple zeta values defined by the author. In this formula, periodlike functions introduced by Lewis and Zagier appear as a counterpart of Eisenstein series.

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arXiv2024

A new deformation of multiple zeta value

We introduce a new deformation of multiple zeta value (MZV). It has one parameter $ω$ satisfying $0<ω<2$ and recovers MZV in the limit as $ω\to +0$. It is defined in the same algebraic framework as a $q$-analogue of multiple zeta value ($q$MZV) by using a multiple integral. We prove that our deformed multiple zeta value satisfies the double shuffle relations which are satisfied by $q$MZVs. We also prove the extended double Ohno relations, which are proved for ($q$)MZVs by Hirose, Sato and Seki, by using a multiple integral whose integrand contains the hyperbolic gamma function due to Ruijsenaars.

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arXiv2023

A q-analogue of symmetric multiple zeta value

We construct a q-analogue of truncated version of symmetric multiple zeta values which satisfies the double shuffle relation. Using it, we define a q-analogue of symmetric multiple zeta values and see that it satisfies many of the same relations as symmetric multiple zeta values, which are the inverse relation and a part of the double shuffle relation and the Ohno-type relation.

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arXiv2022

Supercongruences of multiple harmonic $q$-sums and generalized finite/symmetric multiple zeta values

The Kaneko--Zagier conjecture describes a correspondence between finite multiple zeta values and symmetric multiple zeta values. Its refined version has been established by Jarossay, Rosen and Ono--Seki--Yamamoto. In this paper, we explicate these conjectures through studies of multiple harmonic $q$-sums. We show that the (generalized) finite/symmetric multiple zeta value are obtained by taking an algebraic/analytic limit of multiple harmonic $q$-sums. As applications, new proofs of reversal, duality and cyclic sum formulas for the generalized finite/symmetric multiple zeta values are given.

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arXiv2020

On a weighted sum of multiple T-values of fixed weight and depth

The multiple T-value, which is a variant of multiple zeta value of level two, is introduced by Kaneko and Tsumura. We show that the generating function of a weighted sum of the multiple T-values of fixed weight and depth is given in terms of the multiple T-values of depth one by solving a differential equation of Heun type.

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arXiv2019

Derivations on the algebra of multiple harmonic q-series and their applications

We introduce derivations on the algebra of multiple harmonic q-series and show that they generate linear relations among the q-series which contain the derivation relations for a q-analogue of multiple zeta values due to Bradley. As a byproduct we obtain Ohno-type relations for finite multiple harmonic q-series at a root of unity.

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arXiv2018

Cyclotomic analogues of finite multiple zeta values

We introduce the notion of finite multiple harmonic q-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetrized multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we obtain families of linear relations among these series which induce linear relations among FMZVs and SMZVs of the same form. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic q-series at a primitive root of unity of sufficiently large degree.

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arXiv2018

Special values of finite multiple harmonic q-series at roots of unity

We study special values of finite multiple harmonic q-series at roots of unity. These objects were recently introduced by the authors and it was shown that they have connections to finite and symmetric multiple zeta values and the Kaneko-Zagier conjecture. In this note we give new explicit evaluations for finite multiple harmonic q-series at roots of unity and prove Ohno-Zagier-type relations for them.

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arXiv2017

On the eigenfunctions for the multi-species q-Boson system

In a previous paper a multi-species version of the q-Boson stochastic particle system is introduced and the eigenfunctions of its backward generator are constructed by using a representation of the Hecke algebra. In this article we prove a formula which expresses the eigenfunctions by means of the q-deformed bosonic operators, which are constructed from the L-operator of higher rank found in the recent work by Garbali, de Gier and Wheeler. The L-operator is obtained from the universal R-matrix of the quantum affine algebra of type A_{r}^{(1)} by the use of the q-oscillator representation. Thus our formula may be regarded as a bridge between two approaches to studying integrable stochastic systems by means of the quantum affine algebra and the affine Hecke algebra.

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arXiv2015

Algebraic construction of multi-species q-Boson system

We construct a stochastic particle system which is a multi-species version of the q-Boson system due to Sasamoto and Wadati. Its transition rate matrix is obtained from a representation of a deformation of the affine Hecke algebra of type GL.

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arXiv2014

A deformation of affine Hecke algebra and integrable stochastic particle system

We introduce a deformation of the affine Hecke algebra of type GL which describes the commutation relations of the divided difference operators found by Lascoux and Schutzenberger and the multiplication operators. Making use of its representation we construct an integrable stochastic particle system. It is a generalization of the q-Boson system due to Sasamoto and Wadati. We also construct eigenfunctions of its generator using the propagation operator. As a result we get the same eigenfunctions for the (q, μ, ν)-Boson process obtained by Povolotsky.

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arXiv2014

A discrete analogue of periodic delta Bose gas and affine Hecke algebra

We consider an eigenvalue problem for a discrete analogue of the Hamiltonian of the non-ideal Bose gas with delta-potentials on a circle. It is a two-parameter deformation of the discrete Hamiltonian for joint moments of the partition function of the O'Connell-Yor semi-discrete polymer. We construct the propagation operator by using integral-reflection operators, which give a representation of the affine Hecke algebra. We also construct eigenfunctions by means of the Bethe ansatz method.

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arXiv2013

The Algebra of a q-Analogue of Multiple Harmonic Series

We introduce an algebra which describes the multiplication structure of a family of q-series containing a q-analogue of multiple zeta values. The double shuffle relations are formulated in our framework. They contain a q-analogue of Hoffman's identity for multiple zeta values. We also discuss the dimension of the space spanned by the linear relations realized in our algebra.

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arXiv2010

Quadratic relations for a q-analogue of multiple zeta values

We obtain a class of quadratic relations for a q-analogue of multiple zeta values (qMZV's). In the limit q->1, it turns into Kawashima's relation for multiple zeta values. As a corollary we find that qMZV's satisfy the linear relation contained in Kawashima's relation. In the proof we make use of a q-analogue of Newton series and Bradley's duality formula for finite multiple harmonic q-series.

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