arXiv subjects

Ce Xu

Publications and source records attributed to Ce Xu.

At least 19 recordsLinked to original sources

arXiv2026

Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers

In this paper, we will study a trivariate extension of the Cauchy numbers of both the first kind (also called Gregory coefficients) and the second kind (also called N\"orlund numbers) via the Laurent expansion of the reciprocal of any positive integer power (which is called the order) of multiple polylogarithms. In the case of logarithm, we will show by the WZ method that for each order $\ell>1$ some Gregory coefficient of order $\ell$ must vanish, in contrast to the fact that all classical Gregory coefficients are nonzero. We also prove in this higher order logarithm case that the sequence is eventually alternating for each fixed order, a property enjoyed by the classical Gregory coefficients. In the most general setting, we conjecture that these new sequences are all eventually positive, which is supported by strong numerical evidence. Finally, we confirm this conjecture in the special case of polylogarithms and double polylogarithms. As a by product, for each zeta value and double zeta value, we find an infinite family of identities expressing its reciprocal as a sum of a rational number and an improper integral.

math.NT↗

arXiv2026

Unramified Motivic Alternating Multiple Mixed Values

Many variants of the multiple zeta values have been studied in recent years. There are a few among them that remain to be real numbers, such as the multiple mixed values defined by the authors as level-two generalizations, which include both Hoffman's multiple $t$-values and Kaneko--Tsumura's multiple $T$-values. A central question is when such a value descends to level one, that is, when it can be expressed as a $\Q$-linear combination of multiple zeta values. We call these values unramified. In this paper, we further consider the alternating version of the above variants and identify five families of unramified alternating multiple mixed values using the descent theory of Brown and Glanois. We conjecture that all unramified truly alternating multiple mixed values are given in this paper.

math.NT↗

arXiv2026

Unramified Motivic Multiple Mixed Values

The multiple mixed values (MMVs) are level two variants of multiple zeta values produced by restricting the summation indices to fixed parity patterns. One particularly interesting problem is to determine exactly when such values are actually in level one, namely, expressible in terms of multiple zeta values. To solve this completely is beyond our current knowledge since it calls for new ideas from transcendental number theory. However, much progress has been made on the motivic level. Previously, using the descent theory developed by Brown et al. we have tackled this problem for a few special classes of MMVs with regular parity patterns among the summation indices, including Hoffman's multiple $t$-values, Kaneko and Tsumura's multiple $T$-values, and our own multiple $S$-values. In this paper, we turn to the general case and determine completely all the unramified motivic MMVs of depth less than four as well as a few families of unbounded depths. At the end of the paper, we will present some general conjectures to describe the unramified MMVs at all depths greater than three.

math.AG↗

arXiv2026

Symmetry Results for Cyclotomic Multiple Hurwitz Zeta Values via Contour Integrals

This paper provides a systematic study of symmetry properties for cyclotomic multiple Hurwitz zeta values with multiple variables and parameters by applying the methods of contour integration and the residue theorem. The main contributions are the derivation of explicit symmetry formulas for cyclotomic multiple (Hurwitz) zeta values, which are obtained directly through analytic residue calculations, without reliance on algebraic regularization. As a concrete application, we deduce analogous symmetry theorems for cyclotomic multiple zeta values and cyclotomic multiple $t$-values. The results extend and complement recent symmetry investigations by Charlton and Hoffman, offering completely explicit and regularization-free formulas in the convergent setting. Moreover, the results of this paper can be used to prove the symmetry conjecture for cyclotomic multiple Hurwitz zeta values with multiple variables and a single parameter. Furthermore, several illustrative corollaries and examples are included, and an open problem concerning possible extensions to other variants of multiple zeta values is posed at the conclusion.

math.NT↗

arXiv2026

Residue Theorem, Regularization and Parity Theorem

In this paper, we employ contour integration and residue calculus to derive explicit parity formulas for (cyclotomic) multiple zeta values (MZVs). A key innovation lies in applying double shuffle regularization to the contour integrals, which leads to two distinct regularized parity formulas-one via shuffle and one via stuffle regularization. Notably, this demonstrates for the first time that the contour integral method can be extended to the regularized setting (including the case $k_r=1$), thereby overcoming a limitation of previous approaches. Our results not only provide explicit parity relations at arbitrary depths but also lay the groundwork for extending this technique to other variants of multiple zeta values.

math.NT↗

arXiv2025

Contour Integration and Cyclotomic Ap\'ery-Like Series Involving Generalized Binomial Coefficients

In this paper, we present a method based on contour integration to investigate a class of cyclotomic parametric Ap\'ery-like series. The general term of such series involves a parametric central binomial coefficient, which is defined via the Gamma function. Using this approach, we express a family of cyclotomic Ap\'ery-like series in terms of multiple polylogarithms, cyclotomic Hurwitz zeta values, Riemann zeta values and $\log(2)$. In particular, we provide several illustrative examples and corollaries, which enable us to recover a number of known results on Ap\'ery-like series. At the same time, we have also left open two questions regarding Ap\'ery-like series. Moreover, by considering integrals of the generating function for Fuss-Catalan numbers, we derive an alternative expression for a classical Ap\'ery-like series. Combining this with known results allows us to establish several identities for multiple polylogarithm functions.

math.NT↗

arXiv2025

On a Class of Berndt-type Integrals and Related Barnes Multiple Zeta Functions

This paper investigates a class of special Berndt-type integral calculations where the integrand contains only hyperbolic cosine functions. The research approach proceeds as follows: Firstly, through contour integration methods, we transform the integral into a Ramanujan-type hyperbolic infinite series. Subsequently, we introduce a $\theta$-parameterized auxiliary function and apply the residue theorem from complex analysis to successfully simplify mixed-type denominators combining hyperbolic cosine and sine terms into a normalized Ramanujan-type hyperbolic infinite series with denominators containing only single hyperbolic function terms. For these simplified hyperbolic infinite series, we combine properties of Jacobi elliptic functions with composite analytical techniques involving Fourier series expansion and Maclaurin series expansion. This ultimately yields an explicit expression as a rational polynomial combination of $\Gamma(1/4)$ and $\pi^{-1/2}$. Notably, this work establishes a connection between the integral and Barnes multiple zeta functions, providing a novel research pathway for solving related problems.

math.NT↗

arXiv2025

On the Proof of the Gen\v{c}ev-Rucki Conjecture for Multiple Ap\'ery-Like Series

In this paper, we employ the theories and techniques of hypergeometric functions to provide two distinct proofs of the conjectured identities involving multiple Ap\'ery-like series with central binomial coefficients and multiple harmonic star sums, as recently proposed by Gen\v{c}ev and Rucki. Furthermore, we establish several more general identities for multiple Ap\'ery-like series. Furthermore, by utilizing the method of iterated integrals, a class of multiple mixed values can be expressed as combinations of the multiple Ap\'ery-like series identities conjectured by Gen\v{c}ev and Rucki and $\zeta(2,\ldots,2)$, thus allowing explicit formulas for these multiple mixed values to be derived in terms of Riemann zeta values.

math.NT↗

arXiv2025

The Parity of Two Types of Cyclotomic Euler Sums via Contour Integrals

In this paper, we employ methods of contour integration and residue calculus to investigate the parity of two classes of cyclotomic Euler-type sums. One class involves products of cyclotomic harmonic numbers, while the other involves products of cyclotomic odd harmonic numbers. We derive explicit formulas for the parity of linear and quadratic cases of these cyclotomic Euler-type sums and provide several illustrative examples. The results for the linear and quadratic cases ensure that we can provide explicit formulas for the parity of cyclotomic multiple $T$-values and cyclotomic multiple $S$-values up to depth three, both of which are even-odd variants of cyclotomic multiple zeta values. Furthermore, we present declarative theorems concerning the parity of these two types of cyclotomic Euler-type sums to arbitrary orders. Additionally, using contour integration techniques, we explore explicit linear and quadratic formulas for these cyclotomic Euler-type sums under more general conditions.

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.

math.NT↗

arXiv2025

Contour Integrations and Parity Results of Cyclotomic Euler Sums and Multiple Polylogarithm Function

In this paper, we define extended trigonometric functions via series and employ the method of contour integration to investigate the parity of certain cyclotomic Euler sums and multiple polylogarithm function. We can provide the statement of parity results for cyclotomic Euler sums of arbitrary order, explicit formulas for the parity of cyclotomic linear and quadratic Euler sums, as well as some formulas for the parity of cyclotomic cubic Euler sums and multiple polylogarithms. As a direct corollary, we derive known formulas concerning the parity of classical Euler sums and alternating Euler sums.

math.NT↗

arXiv2025

A Family of Berndt-Type Integrals and Associated Barnes Multiple Zeta Functions

In this paper, we focus on calculating a specific class of Berndt integrals, which exclusively involves (hyperbolic) cosine functions. Initially, this integral is transformed into a Ramanujan-type hyperbolic (infinite) sum via contour integration. Subsequently, a function incorporating theta is defined. By employing the residue theorem, the mixed Ramanujan-type hyperbolic (infinite) sum with both hyperbolic cosine and hyperbolic sine in the denominator is converted into a simpler Ramanujan-type hyperbolic (infinite) sum, which contains only hyperbolic cosine or hyperbolic sine in the denominator. The simpler Ramanujan-type hyperbolic (infinite) sum is then evaluated using Jacobi elliptic functions, Fourier series expansions, and Maclaurin series expansions. Ultimately, the result is expressed as a rational polynomial of Gamma and \sqrt{pi}.Additionally, the integral is related to the Barnes multiple zeta function, which provides an alternative method for its calculation.

math-ph↗

arXiv2025

New Proofs of the Explicit Formulas of Arakawa--Kaneko Zeta Values and Kaneko--Tsumura $\eta$- and $\psi$- Values

In this paper, we establish some new identities of integrals involving multiple polylogarithm functions and their level two analogues in terms of Hurwitz-type multiple zeta (star) values. Using these identities, we provide new proofs of the explicit formulas of Arakawa--Kaneko zeta values, Kaneko--Tsumura $\eta$- and $\psi$-values, and also give a formula for double $T$-values.

math.NT↗

arXiv2024

Mneimneh-type Binomial Sums of Multiple Harmonic-type Sums

In this paper, we establish some expressions of Mneimneh-type binomial sums involving multiple harmonic-type sums in terms of finite sums of Stirling numbers, Bell numbers and some related variables. In particular, we present some new formulas of Mneimneh-type binomial sums involving generalized (alternating) harmonic numbers. Further, we establish a new identity relating the multiple zeta star values $\zeta^\star(m+2,\{1\}_{r-1})$ and specific multiple polylogarithms by applying the Toeplitz principle. Furthermore, we present some interesting consequences and illustrative examples.

math.NT↗

arXiv2024

General Mneimneh-type Binomial Sum involving Harmonic Numbers

Recently, Mneimneh proved the remarkable identity \begin{align*} \sum_{k=0}^n H_k\binom{n}{k} p^k(1-p)^{n-k}=\sum_{i=1}^n \frac{1-(1-p)^i}{i}\quad (p\in [0,1]) \end{align*} as the main result of a 2023 \emph{Discrete Mathematics} paper, where $H_k:=\sum\nolimits_{i=1}^k 1/i$ is the classical $k$-th harmonic number. Thereafter, Campbell provided several other proofs of Mneimneh's formula as above in a note published in \emph{Discrete Mathematics} in 2023. Moreover, Campbell also considered how Mneimneh's identity may be proved and generalized using the \emph{Mathematica package Sigma}. In particular, he found the generalized Mneimneh's identity \begin{align*} \sum_{k=0}^n x^k y^{n-k} \binom{n}{k}H_k =(x+y)^n \left(H_n-\sum_{i=1}^n \frac{y^i (x+y)^{-i}}{i}\right). \end{align*} In this paper, we will prove a more generalization of Mneimneh's identity involving Bell numbers and some Mneimneh-type identities involving (alternating) harmonic numbers by using a few results of our previous papers.

math.NT↗

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 $\Gamma^4(1/4)$ and $\pi^{-1}$ with rational coefficients, where $\Gamma$ is the Euler gamma function. At the end of the paper, we provide some conjectures of general Berndt-type integrals.

math.NT↗

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.

math.CA↗

arXiv2023

Accelerating Split Federated Learning over Wireless Communication Networks

The development of artificial intelligence (AI) provides opportunities for the promotion of deep neural network (DNN)-based applications. However, the large amount of parameters and computational complexity of DNN makes it difficult to deploy it on edge devices which are resource-constrained. An efficient method to address this challenge is model partition/splitting, in which DNN is divided into two parts which are deployed on device and server respectively for co-training or co-inference. In this paper, we consider a split federated learning (SFL) framework that combines the parallel model training mechanism of federated learning (FL) and the model splitting structure of split learning (SL). We consider a practical scenario of heterogeneous devices with individual split points of DNN. We formulate a joint problem of split point selection and bandwidth allocation to minimize the system latency. By using alternating optimization, we decompose the problem into two sub-problems and solve them optimally. Experiment results demonstrate the superiority of our work in latency reduction and accuracy improvement.

cs.LG↗