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Takumi Maesaka

Publications and source records attributed to Takumi Maesaka.

9 recordsLinked to original sources
arXiv2026

Double basic hypergeometric sums via a regularized Jackson $q$-integral

Using the Andrews--Askey integral, we derive a holomorphic extension formula for a Jackson $q$-integral. This formula allows identities established for terminating specializations to be continued to the nonterminating case. Our first application yields two companion binomial-type double-sum formulas. The second formula contains, as special cases, the nonterminating Sears--Carlitz transformation of Gasper and Rahman and Rahman's generating function for the Askey--Wilson polynomials. Applying the same method to the $q$-Chu--Vandermonde and Rogers' ${}_6ϕ_5$ summations, we obtain two further integral representations for double sums of basic hypergeometric type. A balanced ${}_4ϕ_3$ specialization of the latter identity is equivalent, via Sears' transformation, to the double-series transformation of Ismail, Rahman, and Suslov.

math.CO↗
arXiv2026

An analogue of Hirose's relation for finite multiple harmonic $q$-series at roots of unity

Recently, Hirose proved an analogue of the linear part of Kawashima's relations for refined symmetric multiple zeta values. In this paper, we establish an analogue of Hirose's relation for finite multiple harmonic $q$-series at roots of unity. As an application, we prove the cyclic sum conjecture proposed by Kh. Hessami Pilehrood, T. Hessami Pilehrood, and R. Tauraso.

math.NT↗
arXiv2026

Relations and Derivatives of Multiple Eisenstein Series

In this paper, we study multiple Eisenstein series, which build a natural bridge between the theory of multiple zeta values and modular forms. We prove a large family of relations among these series and give an explicit formula for their derivatives. This formula is expressed using the double shuffle structure and the Drop1 operator introduced by Hirose, Maesaka, Seki, and Watanabe. In particular, the space of multiple Eisenstein series is closed under the derivative. Further we construct bi-multiple Eisenstein series, which give a realization of the formal multiple Eisenstein series as holomorphic functions on the upper half-plane, and we prove a conjecture of Okounkov on derivatives of $q$-analogues of multiple zeta values. Based on the derivative formula, we propose a family of linear relations that is conjectured to generate all linear relations among multiple Eisenstein series. Motivated by this conjecture, we introduce a space of formal multiple Eisenstein series and show that it is an $\mathfrak{sl}_2$-algebra.

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arXiv2026

A unified proof of conjectures on the spaces of multiple $q$-zeta values

We prove two conjectures on the spaces generated by multiple $q$-zeta values. More precisely, we show that the spaces $Z_q^{\mathrm{o}}$ and $Z_{q,1}^{\mathrm{o}}$ already generate the larger spaces $Z_q$ and $Z_{q,1}$, respectively. Our result is stronger than the equality of $\mathbb{Q}$-vector spaces: for every generator in the larger spaces, we construct an explicit expression with integer coefficients in terms of the smaller generating families. We first establish these formulas at the finite level, where suitable finite $q$-analogues admit recursive descriptions through generating series, and then pass to the infinite limit.

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arXiv2025

The $\mathbb{Z}$-module of multiple zeta values is generated by ones for indices without ones

We prove that every multiple zeta value is a $\mathbb{Z}$-linear combination of $ζ(k_1,\dots, k_r)$ where $k_i\geq 2$. Our proof also yields an explicit algorithm for such an expansion. The key ingredient is to introduce modified multiple harmonic sums that partially satisfy the relations among multiple zeta values and to determine the structure of the space generated by them.

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arXiv2024

Deriving two dualities simultaneously from a family of identities for multiple harmonic sums

We give a new expression of the multiple harmonic sum, which serves as a refinement of the iterated integral expression of the multiple zeta value, and prove it using the so-called connected sum method. Based on this fact, by taking two kinds of limit operations, we obtain new proofs of both the duality for multiple zeta values and the duality for finite multiple zeta values.

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arXiv2023

Weighted sum formula for variants of half multiple zeta values

We prove some weighted sum formulas for half multiple zeta values, half finite multiple zeta values, and half symmetric multiple zeta values. The key point of our proof is Dougall's identity for the generalized hypergeometric function ${}_{5}F_{4}$. Similar results for interpolated refined symmetric multiple zeta values and half refined symmetric multiple zeta values are also discussed.

math.NT↗
arXiv2021

Multivariable connected sums and multiple polylogarithms

We introduce the multivariable connected sum which is a generalization of Seki-Yamamoto's connected sum and prove the fundamental identity for these sums by series manipulation. This identity yields explicit procedures for evaluating multivariable connected sums and for giving relations among special values of multiple polylogarithms. In particular, our class of relations contains Ohno's relations for multiple polylogarithms.

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