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Jianqiang Zhao

Publications and source records attributed to Jianqiang Zhao.

At least 37 records · Page 2Linked to original sources
arXiv2024

Finite and Symmetric Euler Sums and Finite and Symmetric (Alternating) Multiple $T$-Values

In this paper, we will study finite multiple $T$-values (MTVs) and their alternating versions, which are level two and level four variations of finite multiple zeta values, respectively. We will first provide some structural results for level two finite multiple zeta values (i.e., finite Euler sums) for small weights, guided by the author's previous conjecture that the finite Euler sum space of weight, $w$, is isomorphic to a quotient Euler sum space of weight, $w$. Then, by utilizing some well-known properties of the classical alternating MTVs, we will derive a few important $\Q$-linear relations among the finite alternating MTVs, including the reversal, linear shuffle, and sum relations. We then compute the upper bound for the dimension of the $\Q$-span of finite (alternating) MTVs for some small weights by rigorously using the newly discovered relations, numerically aided by computers.

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arXiv2024

Regularized double shuffle relations of $μ$-multiple Hurwitz zeta values

In this paper we consider a family of multiple Hurwitz zeta values with bi-indices parameterized by $μ$ with $\Ree(μ)>0$. These values are equipped with both the $μ$-stuffle product from their series definition and the shuffle product from their integral expressions. We will give a detailed analysis of the two different products and discuss their regularization when the bi-indices are non-admissible. Our main goal is to prove the comparison theorem relating the two ways of regularization, which is an analog of Ihara, Kaneko, and Zagier's celebrated result concerning the regularization of multiple zeta values. We will see that the comparison formula is $μ$-invariant, that is, it is independent of the parameter $μ$. As applications, we will first provide a few interesting closed formulas for some multi-fold infinite sums that are closely related to the multiple zeta values of level $N$ studied by Yuan and the second author. We will also derive two sum formulas for $μ$-double Hurwitz zeta values which generalize the sum formulas for double zeta values and double zeta star values, respectively, and a weighted sum formula which generalizes the weighted sum formulas for both double zeta values and double $T$-values simultaneously. At the end of the paper, we will propose a problem extending that for multiple zeta values first conjectured by Ihara et al.

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arXiv2024

Finite Multiple Mixed Values

In recent years, a variety of variants of multiple zeta values (MZVs) have been defined and studied. One way to produce these variants is to restrict the indices in the definition of MZVs to some fixed parity pattern, which include Hoffman's multiple $t$-values, Kaneko and Tsumura's multiple $T$-values, and Xu and the author's multiple $S$-values. We have also considered the so-called multiple mixed values by allowing all possible parity patterns and studied a few important relations among these values. In this paper, we will turn to their finite analogs and their symmetric forms, motivated by a deep conjecture of Kaneko and Zagier which relates the finite MZVs and symmetric MZVs, and a generalized version of this conjecture by the author to the Euler sum (i.e., level two) setting. We will present a few important relations among these values such as the stuffle, reversal, and linear shuffle relations. We will also compute explicitly the (conjecturally smallest) generating set in weight one and two cases. In the appendix we tabulate some dimension computation for various sub-spaces of the finite multiple mixed values and propose a conjecture.

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arXiv2024

On Some Unramified Families of Motivic Euler Sums

It is well known that sometimes Euler sums (i.e., alternating multiple zeta values) can be expressed as $\Q$-linear combinations of multiple zeta values (MZVs). In her thesis Glanois presented a criterion for motivic Euler sums to be unramified, namely, expressible as $\Q$-linear combinations of motivic MZVs. By applying this criterion we present a few families of such unramified motivic Euler sums in two groups. In one such group we can further prove the concrete identities relating the motivic Euler sums to the motivic MZVs, determined up to rational multiple of a motivic Riemann zeta value by a result of Brown, under the assumption that the analytic version of such identities hold.

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arXiv2024

General Berndt-Type Integrals and Series Associated with Jacobi Elliptic Functions

In this paper, we prove two structural theorems on the general Berndt-type integrals with the denominator having arbitrary positive degrees by contour integrations involving hyperbolic and trigonometric functions, and hyperbolic sums associated with Jacobi elliptic functions. We first establish explicit relations between these integrals and four classes of hyperbolic sums. Then, using our previous results on hyperbolic series and applying the matrix method from linear algebra, we compute explicitly several general hyperbolic sums and their higher derivatives. These enable us to express two families of general Berndt-type integrals as polynomials in $Γ^4(1/4)$ and $π^{-1}$ with rational coefficients, where $Γ$ is the Euler gamma function. At the end of the paper, we provide some conjectures of general Berndt-type integrals.

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arXiv2024

Mordell--Tornheim Zeta Values, Their Alternating Version, and Their Finite Analogs

The purpose of this paper is two-fold. First, we consider the classical Mordell--Tornheim zeta values and their alternating version. It is well-known that these values can be expressed as rational linear combinations of multiple zeta values (MZVs) and the alternating MZVs, respectively. We show that, however, the spaces generated by these values over the rational numbers are in general much smaller than the MZV space and the alternating MZV space, respectively, which disproves a conjecture of Bachmann, Takeyama and Tasaka. Second, we study supercongruences of some finite sums of multiple integer variables. This kind of congruences is a variation of the so called finite multiple zeta values when the moduli are primes instead of prime powers. In general, these objects can be transformed to finite analogs of the Mordell--Tornheim sums which can be reduced to multiple harmonic sums. This approach not only simplifies the proof of a few previous results but also generalizes some of them. At the end of the paper, we also provide a conjecture supported by strong numerical evidence.

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arXiv2023

Alternating Apéry-Type Series and Colored Multiple Zeta Values of Level Eight

Apéry-type (inverse) binomial series have appeared prominently in the calculations of Feynman integrals in recent years. In our previous work, we showed that a few large classes of the non-alternating Apéry-type (inverse) central binomial series can be evaluated using colored multiple zeta values of level four (i.e., special values of multiple polylogarithms at fourth roots of unity) by expressing them in terms of iterated integrals. In this sequel, we shall prove that for several classes of the alternating versions we need to raise the level to eight. Our main idea is to adopt hyperbolic trigonometric 1-forms to replace the ordinary trigonometric ones used in the non-alternating setting.

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arXiv2023

Berndt-Type Integrals of Order Three and Series Associated with Jacobi Elliptic Functions

In this paper, we first establish explicit evaluations of six classes of hyperbolic sums by special values of the Gamma function by using the tools of the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions developed in our previous paper. Then, using the method of contour integrations involving hyperbolic and trigonometric functions, we establish explicit evaluations of two families of Berndt-type integrals of order three by special values of the Gamma function. Furthermore, we present some interesting consequences and illustrative examples.

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arXiv2023

Weighted and Restricted Sum Formulas of Euler Sums

One of the most interesting formulas for multiple zeta values is the sum formula proved by Granville and Zagier independently in 1990s. Many variations and generalizations of it have been found since then. In this paper, we will provide a uniform approach to proving these results using the regularized double shuffle relations of Euler sums. We summarize these formulas including many new ones with three tables in the Appendix at the end of the paper.

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arXiv2023

On a problem involving the squares of odd harmonic numbers

We introduce a full solution to a problem considered by Wang and Chu concerning series involving the squares of finite sums of the form $1 + \frac{1}{3}+ \cdots + \frac{1}{2n-1}$. Our proof involves techniques from the theory of colored multiple zeta values.

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arXiv2022

On two conjectures of Sun concerning Apéry-like series

In this paper, we shall prove two conjectures of Z.-W. Sun concerning Apéry-like series. One of the series is alternating whereas the other one is not. Our main strategy is to convert the series (resp.~the alternating series) to log-sine-cosine (resp.~log-sinh-cosh) integrals. Then we express all these integrals in terms of single-valued Bloch-Wigner-Ramakrishnan-Wojtkowiak-Zagier polylogarithms. The conjectures then follow from a few highly non-trivial functional equations of the polylogarithms of weight $3$ and $4$.

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arXiv2022

Explicit Evaluation of Euler-Apéry Type Multiple Zeta Star Values and Multiple $t$-Star Values

In this paper we establish several recurrence relations about Euler-Apéry type multiple zeta star values and a parametric variant of it by using the method of iterated integrals. Then using the formulas obtained, we find the explicit evaluations for some specific Euler-Apéry type multiple zeta star values and one of its parametric variant, and Euler-Apéry type multiple $t$-star values.

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arXiv2022

On $N$-Unital Functions

Let $N$ be a positive integer. We say a non-constant rational function $U(x)\in{\mathbb C}(x)$ is $N$-\emph{unital} if all the zeros and poles of both $U(x)$ and $1-U(x)$ are either 0 or $N$-th roots of unity. These functions are called admissible functions by Au in a recent paper arXiv:2007.03957 and used to study some central binomial series of Apéry type via their iterated integral expression related to multiple polylogrithms and colored multiple zeta values. In this paper we determine the complete set of these functions for $N\le 4$ by elementary method, and briefly study some cases at level 5.

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arXiv2022

Apéry-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, I

In this paper, we shall study Aéry-type series in which the central binomial coefficient appears as part of the summand. Let $b_n=4^n/\binom{2n}{n}$. Let $s_1,\dots,s_d$ be positive integers with $s_1\ge 2$. We consider the series \begin{align*} \sum_{n_1>\cdots>n_d>0} \frac{b_{n_1}}{n_1^{s_1}\cdots n_d^{s_d}} \end{align*} and the variants with some or all indices $n_j$ replaced by $2n_j\pm 1$ and some or all "$>$" replaced by "$\ge$", provided the series are defined. We can also replace $b_{n_1}$ by its square in the above series when $s_1\ge 3$. The main result is that all such series are $\mathbb{Q}$-linear combinations of the real and/or the imaginary parts of some colored multiple zeta values of level 4, i.e., multiple polylogarithms evaluated at 4th roots of unity.

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arXiv2022

Explicit Relations of Some Variants of Convoluted Multiple Zeta Values

Kaneko and Yamamoto introduced a convoluted variant of multiple zeta values (MVZs) around 2016. In this paper, we will first establish some explicit formulas involving these values and their alternating version by using iterated integrals, which enable us to derive some explicit relations of the multiple polylogarithm (MPL) functions. Next, we define convoluted multiple $t$-values and multiple mixed values (MMVs) as level two analogs of convoluted MZVs, and, similar to convoluted MZVs, use iterated integrals to find some relations of these level two analogs. We will then consider the parametric MPLs and the parametric multiple harmonic (star) sums, and extend the Kaneko-Yamamoto's "integral-series" identity of MZVs to MPLs and MMVs. Finally, we will study multiple integrals of MPLs and MMVs by generalizing Yamamoto's graphical representations to multiple-labeled posets.

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arXiv2022

Apéry-Type Series and Colored Multiple Zeta Values

In this paper, we introduce and study new classes of Apéry-type series involving the multiple $t$-harmonic sums by combining the methods of iterated integral and Fourier--Legendre series expansions, where the multiple $t$-harmonic sums are a variation of multiple harmonic sums in which all the summation indices are restricted to odd numbers only. Our approach also enables us to generalize some old classes of Apéry-type series involving harmonic sums to those with products of harmonic sums and multiple $t$-harmonic sums. In all of these series, the central binomial coefficients appear as $a_n^{\pm 1}$ or $a_n^{\pm 2}$ where $a_n=\binom{2n}{n}/4^n$. We show that every such series can be expressed as either the real or the imaginary part of a $\mathbb Q$-linear combination of colored multiple zeta values of level 4.

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