arXiv subjects

Ce Xu

Publications and source records attributed to Ce Xu.

At least 37 records · Page 2Linked to original sources

arXiv2024

Parametric Apéry-type Series and Hurwitz-type Multiple Zeta Values

In this paper, we extend the main results of a 2024 \emph{Advances in Applied Mathematics} paper \cite{XuZhao2021c} about Apéry-type series involving central binomial coefficients and the multiple ($t-$)harmonic sums to parametric Apéry-type series involving parametric binomial coefficients and Hurwitz-type multiple harmonic (star) sums. In particular, we will establish many explicit relations between parametric Apéry-type series involving one or two parametric binomial coefficients and Hurwitz-type multiple zeta values (with $r$-variables) by using the method of iterated integrals.

math.NT↗

arXiv2024

Reciprocal Hyperbolic Series of Ramanujan Type

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of $z={}_2F_1(1/2,1/2;1;x)$ and $z'=dz/dx$. When a certain parameter in these series is equal to $π$ the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

math.NT↗

arXiv2024

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

In this paper, we evaluate in closed forms two families of infinite integrals containing hyperbolic and trigonometric functions in their integrands. We call them Berndt-type integrals since he initiated the study of similar integrals. We first establish explicit evaluations of four classes of hyperbolic sums by special values of the Gamma function, by two completely different approaches, which extend those sums considered by Ramanujan and Zucker previously. We discover the first by refining two results of Ramanujan concerning some $q$-series. For the second we compare both the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions. Next, by contour integrations we convert two families of Berndt-type integrals to the above hyperbolic sums, all of which can be evaluated in closed forms. We then discover explicit formulas for one of the two families. Throughout the paper we present many examples which enable us to formulate a conjectural explicit formula for the other family of the Berndt-type integrals at the end.

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 $ζ^\star(m+2,\{1\}_{r-1})$ 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

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.

math.NT↗

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.

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↗

arXiv2023

On two conjectural series involving Riemann zeta function

Riemann zeta function is important in a lot of branches of number theory. With the help of the operator method and several transformation formulas for hypergeometric series, we prove four series involving Riemann zeta function. Two of them are series expansions for $ζ(7)$ and $ζ(3)^2$ recently conjectured by Z.-W. Sun.

math.CO↗

arXiv2023

On some conjectural series containing harmonic numbers of 3-order

Harmonic numbers are important in a lot of branches of number theory. By means of the derivative operator, the integral operator, and several summation and transformation formulas for hypergeometric series, we prove four series containing harmonic numbers of 3-order. Three of them are conjectures which were recently proposed by Z.-W. Sun.

math.CO↗

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.

math.NT↗

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 $4$.

math.NT↗

arXiv2022

Some results on Arakawa-Kaneko, Kaneko-Tsumura functions and related functions

Recently, the level two analogue of multiple polylogarithm function ${\rm A}(k_1,\ldots,k_r;z)$ and Arakawa-Kaneko zeta function $ψ(k_1,\ldots,k_r;s)$ were introduced by M. Kaneko and H. Tsumura, for $k_1,\ldots,k_r \in \mathbb{Z}_{\ge 1}$ . In this paper, we investigate some of their special relations. In particular, we prove some explicit forms of ${\rm A}(k_1,\ldots,k_r;z)$ and $ψ(k_1,\ldots,k_r;s)$. Also, we introduce a level $m$ anlogue of the Arakawa-Kaneko zeta functions.

math.NT↗

arXiv2022

Some Evaluations of Parametric Euler Type Sums of Harmonic Numbers

We establish some identities of Euler related sums. By using these identities, we discuss the closed form representations of sums of harmonic numbers and reciprocal parametric binomial coefficients through parametric harmonic numbers, shifted harmonic numbers and Riemann zeta function with positive integer arguments. In particular we investigate products of quadratic and cubic harmonic numbers and reciprocal parametric binomial coefficients. Some illustrative special cases as well as immediate consequences of the main results are also considered.

math.NT↗

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.

math.NT↗

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.

math.NT↗