arXiv subjects

Ku-Yu Fan

Publications and source records attributed to Ku-Yu Fan.

5 recordsLinked to original sources

arXiv2026

Regularised double shuffle relations for planar Arborified Zeta Values

We endow spaces of decorated planar rooted trees with new dendriform and tridendrifrom algebra structures and provide their combinatorial description. We then show that the planar counterparts of Arborified Zeta Values are algebra morphisms for these shuffle and quasi-shuffle products of planar rooted trees. We also prove an arborified version of Hoffman's regularisation relation for Arborified Zeta Values. We conjecture that those give every rational relation between Arborified Zeta Values and show that this conjecture implies the regularised double shuffle conjecture for Multiple Zeta Values.

math.NT↗

arXiv2026

$p$-adic multiple $L$-functions and twisted multiple Bernoulli numbers

We compute the special values ($p$MLFVs) of the $p$-adic multiple $L$-functions introduced by Furusho, Komori, Matsumoto, and Tsumura at tuples of positive integers. Furusho and Jarossay show that the special values can be expressed as an infinite sum of cyclotomic multiple harmonic values (CMHVs) with coefficients given by cyclotomic multiple Bernoulli numbers (CMBNs). We provide an explicit formula for CMBNs in terms of twisted multiple Bernoulli numbers (TMBNs), which are special values of generalized Euler-Zagier-Lerch type complex multiple zeta functions at tuples of non-positive integers. As a result, we obtain that these $p$MLFVs can be expressed as infinite sums of CMHVs, with coefficients given by the special values of the complex functions at tuples of non-positive integers.

math.NT↗

arXiv2026

$p$-adic multiple zeta values of integer indices

This paper concerns the $p$-adic multiple zeta values of integer indices that may contain zero or negative components. We introduce the admissibility and regularizability conditions for integer indices. We define the $p$-adic multiple zeta values associated with admissible integer indices to be finite rational linear combinations of $p$-adic multiple zeta values associated with admissible positive integer indices. We prove that the double shuffle relations, that is, the shuffle and stuffle product formulas, both hold for the values.

math.NT↗

arXiv2025

A map between arborifications of multiple zeta values

Arborified multiple zeta values are a generalization of multiple zeta values associated with rooted trees. There are two types of decorated rooted trees, corresponding respectively to the series and the integral expressions. Manchon introduces the contracting arborification (resp. the simple arborification), which is maps from the BCK Hopf algebras of the decorated rooted trees corresponding to the series expression (resp. the integral expression) to the non-commutative polynomial algebras of the set $\mathbb{N}$ (resp. the set $\{0,1\}$). There is a natural map between the two non-commutative polynomial algebras. Manchon posed the question of finding a natural map between the two BCK Hopf algebras that would make the diagram commutative. In this paper, we consider planar rooted trees and use a recursive method to construct such a map between the two BCK Hopf algebras, making the diagram commutative.

math.NT↗

arXiv2024

Coproduct Formula for Motivic Version of Yamamoto's Integral

Goncharov proved an explicit formula for the coproduct in the Hopf algebra of motivic iterated integrals. Yamamoto introduced Yamamoto's integral which generalizes iterated integrals and gave a new integral expression for multiple zeta star values using Yamamoto's integral. In this paper, we consider the motivic version of Yamamoto's integral and generalize Goncharov's coproduct formula to those motivic integrals. As an example, we will compute the coproduct of a certain type of Schur multiple zeta values.

math.NT↗