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Zhenlu Wang

Publications and source records attributed to Zhenlu Wang.

9 recordsLinked to original sources
arXiv2026

A proof of a weighted sum conjecture for finite multiple zeta values of level two

M. Kaneko, T. Murakami and A. Yoshihara introduced finite multiple zeta values of level two and conjectured a weighted sum formula for indices whose components belong to $\{1,2\}$. In this paper, we use generating functions and linear recurrence relations to prove the conjecture. More precisely, we reduce the problem to a polynomial identity and solve the resulting second-order difference equation by identifying its specialized solutions with $_4F_3$ hypergeometric polynomials of Racah-type.

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arXiv2026

Weighted sum formulas for finite multiple mixed values

In this paper, we employ the iterated integral expression of multiple polylogarithms to establish a weighted sum formula for finite multiple mixed values. As applications, we derive various relations among level-two variants of finite multiple zeta values.

math.NT↗
arXiv2025

Contour Integrations and Parity Results of Cyclotomic Euler $T$-Sums and Multiple $t$-Values

We will employ the method of contour integration to investigate the parity results of non-embedded cyclotomic multiple $t$-values, which we refer to as cyclotomic Euler $T$-sums. We can provide explicit parity formulas for the linear and quadratic cases of cyclotomic Euler $T$-sums, as well as state a parity theorem for the general case. We also present illustrative examples and corollaries. From this, some parity results for classical cyclotomic multiple $t$-values can be derived. Furthermore, we present several general formulas for cyclotomic Euler $T$-sums with denominators involving arbitrary rational polynomials through residue computations. By evaluating these polynomials and computing residues, many other formulas analogous to cyclotomic Euler $T$-sums can be derived. In particular, we also obtain certain parity results for the cyclotomic versions of multiple $T$-values as defined by Kaneko and Tsumura. Finally, we propose some conjectures and questions regarding the parity of cyclotomic multiple $t$-values and cyclotomic multiple $T$-values.

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arXiv2025

Interpolated multiple $t$-values of general level with fixed weight, depth and height

In this paper, we introduce the interpolated multiple $t$-values of general level and represent a generating function for sums of interpolated multiple $t$-values of general level with fixed weight, depth, and height in terms of a generalized hypergeometric function $_3F_2$ evaluated at $1$. Furthermore, we explore several special cases of our results. The theorems presented in this paper extend earlier results on multiple zeta values and multiple $t$-values of general level.

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arXiv2022

Relations of multiple $t$-values of general level

We study the relations of multiple $t$-values of general level. The generating function of sums of multiple $t$-(star) values of level $N$ with fixed weight, depth and height is represented by the generalized hypergeometric function $_3F_2$, which generalizes the results for multiple zeta(-star) values and multiple $t$-(star) values. As applications, we obtain formulas for the generating functions of sums of multiple $t$-(star) values of level $N$ with height one and maximal height and a weighted sum formula for sums of multiple $t$-(star) values of level $N$ with fixed weight and depth. Using the stuffle algebra, we also get the symmetric sum formulas and Hoffman's restricted sum formulas for multiple $t$-(star) values of level $N$. Some evaluations of multiple $t$-star values of level $2$ with one-two-three indices are given.

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arXiv2022

Integrality and some evaluations of odd multiple harmonic sums

In 2015, S. Hong and C. Wang proved that none of the elementary symmetric functions of $1,1/3,\ldots,1/(2n-1)$ is an integer when $n\geq 2$. In 2017, Kh. Pilehrood, T. Pilehrood and R. Tauraso proved that the multiple harmonic sums $H_n(s_1,\ldots,s_r)$ are never integers with exceptions of $H_1(s_1)=1$ and $H_3(1,1)=1$. They also proved that the multiple harmonic star sums are never integers when $n\geq 2$. In this paper, we consider the odd multiple harmonic sums and the odd multiple harmonic star sums and show that none of these sums is an integer with exception of the trivial case. Besides, we give evaluations of the odd (alternating) multiple harmonic sums with depth one.

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arXiv2021

A note on the sum of finite multiple harmonic $q$-series on $r\text{-}(r+1)$ indices

We study the sum of the finite multiple harmonic $q$-series on $r\text{-}(r+1)$ indices at roots of unity with $r=1,2,3$. And we give the equivalent conditions of two conjectures regarding cyclic sums of finite multiple harmonic $q$-series on $1\text{-}2\text{-}3$ indices at roots of unity, posed recently by Kh. Pilehrood, T. Pilehrood and R. Tauraso.

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arXiv2021

Weighted sum formula of multiple $L$-values and its applications

In this paper, we study the multiple $L$-values and the multiple zeta values of level $N$. We set up the algebraic framework for the double shuffle relations of the multiple zeta values of level $N$. Using the regularized double shuffle relations of multiple $L$-values, we give a sum formula and a weighted sum formula of multiple $L$-values. As applications, we give sum formulas and weighted sum formulas of double zeta values of level $2$ and $3$.

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