arXiv subjects
Publications and source records attributed to Minoru Hirose.
Nakasuji, Phuksuwan, and Yamasaki defined the Schur multiple zeta values and gave iterated integral expressions of the Schur multiple zeta values of the ribbon type. This paper generalizes their integral expressions to the ones of more general Schur multiple zeta values having constant entries on the diagonals. Furthermore, we also discuss the duality relations for Schur multiple zeta values obtained from the integral expressions.
We show that the level 2 case of the cyclotomic Grothendieck-Teichmüller groups introduced by Enriquez coincides with the motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$.
We define the interpolated polynomial multiple zeta values as a generalization of all of multiple zeta values, multiple zeta-star values, interpolated multiple zeta values, symmetric multiple zeta values, and polynomial multiple zeta values. We then compute the generating function of the sum of interpolated polynomial multiple zeta values of fixed weight, depth, and height.
In 1998, Borwein, Bradley, Broadhurst and Lisoněk posed two families of conjectural identities among multiple zeta values, later generalized by Charlton using his alternating block notation. In this paper, we prove a new class of identities among multiple zeta values that simultaneously resolve and generalize these conjectures.
We consider the symmetric multiple zeta values in $\mathcal{S}_m$ without modulo $π^2$ reduction for indices in which $1$ and $3$ appear alternately. We investigate those values that can be expressed as a polynomial of the Riemann zeta values, and give a conjecturally complete list of explicit formulas for such values.
In this paper we introduce confluence relations for motivic Euler sums (also called alternating multiple zeta values) and show that all linear relations among motivic Euler sums are exhausted by the confluence relations. This determines all automorphisms of the de Rham fundamental groupoid of $\mathbb{P}^{1}\setminus\{0,\pm1,\infty\}$ coming from the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$. Moreover, we also discuss other applications of the confluence relations such as an explicit $\mathbb{Q}$-linear expansion of a given motivic Euler sum by their basis and $2$-adic integrality of the coefficients in the expansion.
Parametrized multiple series are generalizations of the multiple zeta values introduced by Igarashi. In this work, we completely determine all the linear relations among these parameterized multiple series. Specifically, we prove the following two statements: the linear part of the Kawashima relation for multiple zeta values can be generalized to the parametrized multiple series; any linear relations among the parametrized multiple series can be written as a linear combination of the linear part of the Kawashima relation.
The sum formula is a well known relation in the field of the multiple zeta values. In this paper, we present its generalization for the Euler-Zagier multiple zeta function.
Recently, Bényi and the second author introduced two combinatorial interpretations for symmetrized poly-Bernoulli polynomials. In the present study, we construct bijections between these combinatorial objects. We also define various combinatorial polynomials and prove that all of these polynomials coincide with symmetrized poly-Bernoulli polynomials.
In this paper, we investigate linear relations among regularized motivic iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ of depth two, which we call regularized motivic double zeta values. Some mysterious connections between motivic multiple zeta values and modular forms are known, e.g. Gangl--Kaneko--Zagier relation for the totally odd double zeta values and Ihara--Takao relation for the depth graded motivic Lie algebra. In this paper, we investigate so-called non-admissible cases and give many new Gangl--Kaneko--Zagier type and Ihara--Takao type relations for regularized motivic double zeta values. Specifically, we construct linear relations among a certain family of regularized motivic double zeta values from odd period polynomials of modular forms for the unique index two congruence subgroup of the full modular group. This gives the first non trivial example of a construction of the relations among multiple zeta values (or their analogues) from modular forms for a congruence subgroup other than the ${\rm SL}_{2}(\mathbb{Z})$.
The Ohno relation for multiple zeta values can be formulated as saying that a certain operator, defined for indices, is invariant under taking duals. In this paper, we generalize the Ohno relation to regularized multiple zeta values by showing that, although the suitably generalized operator is not invariant under taking duals, the relation between its values at an index and at its dual index can be written explicitly in terms of the gamma function.
We prove the Ohno-type relation for the interpolated multiple zeta values, which was introduced first by Yamamoto. Same type results for finite multiple zeta values are also given. Moreover, these relations give the sum formula for interpolated multiple zeta values and interpolated $\mathcal{F}$-multiple zeta values, which were proved by Yamamoto and Seki, respectively.
The sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta values. We also obtain the sum formula for polynomial multiple zeta(-star) values in terms of generating functions, simultaneously generalizing the sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values.
Ohno's relation is a generalization of both the sum formula and the duality formula for multiple zeta values. Oyama gave a similar relation for finite multiple zeta values, defined by Kaneko and Zagier. In this paper, we prove relations of similar nature for both multiple zeta-star values and finite multiple zeta-star values. Our proof for multiple zeta-star values uses the linear part of Kawashima's relation.
In this paper, we introduce a new connector which generalizes the connector found by the third author and Yamamoto. The new connector gives a direct proof of the double Ohno relation recently proved by the first author, the second author, Murahara, and Onozuka. Furthermore, we obtain a simultaneous generalization of the ($q$-)Ohno relation and the ($q$-)double Ohno relation.
The $t$-adic symmetric multiple zeta values were defined Jarossay, which have been studied as a real analogue of $\boldsymbol{p}$-adic finite multiple zeta values. In this paper, we consider the star analogues based on several regularization processes of multiple zeta-star values: harmonic regularization, shuffle regularization, and Kaneko-Yamamoto's type regularization. We also present the cyclic sum formula for $t$-adic symmetric multiple zeta(-star) values, which is the counterpart of that for $\boldsymbol{p}$-adic finite multiple zeta(-star) values obtained by Kawasaki. The proof uses our new relationship that connects the cyclic sum formula for $t$-adic symmetric multiple zeta-star values and that for the multiple zeta-star values.
Ohno's relation is a well-known family of relations among multiple zeta values, which can naturally be regarded as a type of duality for a certain power series which we call an Ohno sum. In this paper, we investigate $\mathbb{Q}$-linear relations among Ohno sums which are not contained in Ohno's relation. We prove two new families of such relations, and pose several further conjectural families of such relations.
In this paper, we study specific families of multiple zeta values which closely relate to the linear part of Kawashima's relation. We obtain an explicit basis of these families, and investigate their interpolations to complex functions. As a corollary of our main results, we also see that the duality formula and the derivation relation are deduced from the linear part of Kawashima's relation.