Sergey Zlobin
We study properties of coefficients of a linear form, originating from a multiple integral. As a corollary, we prove Vasilyev's conjecture, connected with the problem of irrationality of the Riemann zeta function at odd integers.
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In this paper, we give explicit evaluation for some infinite series involving generalized (alternating) harmonic numbers. In addition, some formulas for generalized (alternating) harmonic numbers will also be derived.
We study values of generalized polylogarithms at various points and relationships among them. Polylogarithms of small weight at the points 1/2 and -1 are completely investigated. We formulate a conjecture about the structure of the linear space generated by values of generalized polylogarithms.
A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of one--dimensional hyperfunctions are also constructed.
In this paper, we study a family of single variable integral representations for some products of $ζ(2n+1)$, where $ζ(z)$ is Riemann zeta function and $n$ is positive integer. Such representation involves the integral $Lz(a,b):=\frac{1}{(a-1)!b!}\int_{0}^{1}\log^a (t)\log^b (1-t)dt/t$ with positive integers $a,b$, which is related to Nielsen's generalized polylogarithms. By analyzing the related partition problem, we discuss the structure of such integral representation, especially the condition of expressing products of $ζ(2n+1)$ by finite $\mathbb{Q}(π)$-linear combination of $Lz(a,b)$.
In this article, we derive a system of functional relations called the generalized harmonic product relations for hyperlogarithms on the moduli space ${\mathcal M}_{0,5}$ and show that the relations contain the harmonic product of multiple polylogarithms. The generalized harmonic product relations are equivalent to the relations which come from two decompositions of the fundamental solution normalized at the origin of the KZ equation on ${\mathcal M}_{0,5}$.
We introduce a generalization $£_{d}^{(α)}(X)$ of the finite polylogarithms $£_{d}^{(0)}(X)=£_d(X)=\sum_{k=1}^{p-1}X^k/k^d$, in characteristic $p$, which depends on a parameter $α$. The special case $£_{1}^{(α)}(X)$ was previously investigated by the authors as the inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential which is instrumental in a {\em grading switching} technique for non-associative algebras. Here we extend such generalization to $£_{d}^{(α)}(X)$ in a natural manner, and study some properties satisfied by those polynomials. In particular…
For a given rational number $x$ and an integer $s\geq 1$, let us consider a generalized polylogarithmic function, often called the Lerch function, defined by $$Φ_{s}(x,z)= \sum_{k=0}^{\infty}\frac{z^{k+1}}{(k+x+1)^s}\enspace.$$ We prove the linear independence over any number field $K$ of the numbers $1$ and $Φ_{s_j}(x_j,α_i)$ with any choice of distinct shifts $x_1,\ldots, x_d$ with $0\le x_1<\ldots<x_d<1$, as well as any choice of depths $1\leq s_1\leq r_1,\ldots, 1\leq s_d\leq r_d$, at distinct algebraic numbers $α_1,\ldots,α_m\in K$ subject to a metric condition. As is usual in the theory,…
We describe a solution of the Gauss hypergeometric equation, $F(α,β,γ;z)$ by power series in paramaters $α,β,γ$ whose coefficients are $\Z$ linear combinations of multiple polylogarithms. And using the representation and connection reletions of solutions of the hypergeometric equation, we show some relations of multiple zeta values.
We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients $\binom{2k}{k}$.
Laurent type expansion of multiple polylogarithms at integer points
本文通过Laurent展开研究多重polylogarithms在整数点的局部行为，揭示其正则化系数与相关整数点的关系。
Recently, several people study finite multiple zeta values (FMZVs) and finite polylogarithms (FPs). In this paper, we introduce finite multiple polylogarithms (FMPs), which are natural generalizations of FMZVs and FPs, and we establish functional equations of FMPs. As applications of these functional equations, we calculate special values of FMPs containing generalizations of congruences obtained by Meštrović, Z. W. Sun, Z. W. Sun-L. L. Zhao, and Tauraso-J. Zhao. We show supercongruences for certain generalized Bernoulli numbers and the Bernoulli numbers as an appendix.
We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating function and sets of generalized coefficients satisfying a non-triangular recurrence relation in two variables. The generalized transformation coefficients share a number of analogous properties with the Stirling numbers of the second kind and the known harmonic number expansions of the unsigned Stirling numbers of the first kind. We prove a number of propertie…
Our main aim in this paper is to give a foundation of the theory of $p$-adic multiple zeta values. We introduce (one variable) $p$-adic multiple polylogarithms by Coleman's $p$-adic iterated integration theory. We define $p$-adic multiple zeta values to be special values of $p$-adic multiple polylogarithms. We consider the (formal) $p$-adic KZ equation and introduce the $p$-adic Drinfel'd associator by using certain two fundamental solutions of the $p$-adic KZ equation. We show that our $p$-adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the $p$-adic KZ…
Closed-form evaluations of log-sine integrals and Apéry-like sums in terms of polylogarithms
本文提出一种基于多对数生成函数与正弦积分恒等式的新方法，系统求值广义对数正弦积分，得到闭式表达式，并推广出众多类阿佩里公式及双曲类比，扩展了已有结果。
Pan-Xu conjecture and reduction formulas for polylogarithms
本文研究了多重调和星值参数变体的类Mneimneh和，推广并解决了Pan-Xu猜想，推导出特定多重对数的深度约化公式及多重zeta星值的新表示。
The goal of the paper is to give a systematic way to numerically evaluate the generating function of a periodic multiple polylogarithm using a Chen-Fliess series with a rational generating series. The idea is to realize the corresponding Chen-Fliess series as a bilinear dynamical system. A standard form for such a realization is given. The method is also generalized to the case where the multiple polylogarithm has non-periodic components. This allows one, for instance, to numerically validate the Hoffman conjecture. Finally, a setting in terms of dendriform algebras is provided.
Asymptotic expansions of integrals and Nielsen's polylogarithms
本文推导了形式为∫₀¹f(u)(1+q⋅uⁿ)^(w/n)du的积分在n→∞时的渐近展开，将系数与尼尔森广义多项对数关联，并对q=-1时的展开涉及多重ζ值，q=1时的展开涉及交替多重ζ值，同时验证了多项式变换的对称性条件。
Linear independence of values of polylogarithms with periodic coefficients, outside the disk of convergence
本文针对周期系数的多对数函数值，在收敛圆盘外通过解析延拓定义，利用精细Siegel引理和线性无关性判据，给出了其维数的下界，推广了Fischler的结果。
The Galois action on the pro-$\ell$ étale fundamental groupoid of the projective line minus three points with rational base points gives rise to a non-commutative formal power series in two variables with $\ell$-adic coefficients, called the $\ell$-adic Galois associator. In the present paper, we focus on how Landen's functional equation of trilogarithms and its $\ell$-adic Galois analog can be derived algebraically from the $S_3$-symmetry of the projective line minus three points. Twofold proofs of the functional equation will be presented, one is based on Zagier's tensor criterion devised in…
Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers
本文通过多重对数倒数的Laurent展开定义三变量Cauchy数，证明高阶Gregory系数有零点且序列最终交错，并猜想及部分证实其最终正性，同时给出zeta值倒数的新恒等式族。
Iterated Integrals and Multiple Polylogarithm at Algebraic Arguments
本文通过迭代积分与广义多重ζ值理论，揭示了代数参数下多重polylogarithm与cyclotomic 多重ζ值之间的等价关系，并提出Galois下降猜想及Apéry型级数的应用。
In this paper, we establish more identities of generalized multi poly-Euler polynomials with three parameters and obtain a kind of symmetrized generalization of the polynomials. Moreover, generalized multi poly-Bernoulli polynomials are defined using multiple polylogarithm and derive some properties parallel to those of poly-Bernoulli polynomials. These are generalized further using the concept of Hurwitz-Lerch multiple zeta values.
Contour Integration and Cyclotomic Apéry-Like Series Involving Generalized Binomial Coefficients
本文通过轮廓积分方法研究了一类涉及广义二项式系数的 cyclotomic 阿佩里型级数，推导出多重 polylogarithms 等表达式，并恢复了一些已知结果。
In earlier work we studied features of non-holomorphic modular functions associated with Feynman graphs for a conformal scalar field theory on a two-dimensional torus with zero external momenta at all vertices. Such functions, which we will refer to as modular graph functions, arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We here introduce a class of single-valued elliptic multiple polylogarithms, which are defined as elliptic functions associated with Feynman graphs with vanishing external momenta at all but two vertices. These functions depend o…
For each positive characteristic multiple zeta value (defined by Thakur), the first and third authors constructed a $t$-module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value. The objective of this paper is to use the Taylor coefficients of Anderson-Thakur series and $t$-motivic Carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector.
The integral $t$-motivic cohomology and the class module of a (rigid analytically trivial) Anderson $t$-motive were introduced by the first author in [Gaz22b]. This paper is devoted to their determination in the particular case of tensor powers of the Carlitz $t$-motive, namely, the function field counterpart $\underline{A}(n)$ of Tate twists $\mathbb{Z}(n)$. We find out that these modules are in relation with fundamental objects of function field arithmetic: integral $t$-motivic cohomology governs linear relations among Carlitz polylogarithms, its torsion is expressed in terms of the denomina…
An arbitrary-depth reduction theorem for the `convolution' multiple L-values of Euler-Zagier type is proven by an analytic method. To this end, generalized polylogarithms associated to Dirichlet characters are defined. The proof uses the monodromies of these functions and the parities of generating functions of generalized Bernoulli numbers. Some general formulas for resulting reductions in the depth 2 case are provided.
We introduce an iterated integral version of (generalized) log-sine integrals (iterated log-sine integrals) and prove a relation between a multiple polylogarithm and iterated log-sine integrals. We also give a new method for obtaining relations among multiple zeta values, which uses iterated log-sine integrals, and give alternative proofs of several known results related to multiple zeta values and log-sine integrals.
We study analytic properties of multiple zeta-functions of generalized Hurwitz-Lerch type. First, as a special type of them, we consider multiple zeta-functions of generalized Euler-Zagier-Lerch type and investigate their analytic properties which were already announced in our previous paper. Next we give `desingularization' of multiple zeta-functions of generalized Hurwitz-Lerch type, which include those of generalized Euler-Zagier-Lerch type, the Mordell-Tornheim type, and so on. As a result, the desingularized multiple zeta-function turns out to be an entire function and can be expressed as…
In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.
Elliptic modular graph forms, equivariant iterated integrals and single-valued elliptic polylogarithms
通过等变迭代积分和单值椭圆多项式对数构造单值椭圆模形式
Using generalized binomial coefficients with respect to fundamental Lucas sequences we establish congruences that generalize the classical congruence of Wolstenholme and other related stronger congruences.
In this article, we consider weighted sums of generalized polygonal numbers with coefficients $1$ or $2$. We show that for any $m\ge10$, those weighted sums of generalized $m$-gonal numbers represent every non-negative integers if they only represent $1$, $m-4$, and $m-2$. Furthermore, we study representations of sums of four generalized polygonal numbers with coefficients $1$ or $2$.
In this paper we showed that under two assumptions we are able to define interesting functions that we call generalized local coefficients. We showed that in the quasi-split case generalized local coefficients are up to a positive constant the same as Shahidi's local coefficients. We provide a proof that the non quasi-split group $GL_m(D)$, for a central division algebra $D$ satisfies those assumptions. We also showed that generalized local coefficients satisfy nice properties, like the relation to Plancherel measures and multiplicativity inherited by that of intertwining operators. Generalize…
Asymptotic coefficients of multiple zeta functions at the origin and generalized Gregory coefficients
本文研究多重zeta函数在原点的渐近系数，揭示其生成级数为广义Gregory系数，并推广至Hurwitz多重zeta函数，揭示对称性。
Let f be a modular form with complex multiplication. If f has critical slope, then Coleman's classicality theorem implies that there is a p-adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as f. We give a formula for the Fourier coefficiets of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of p-adic overconvergent forms containing f.
Integer relation algorithms can convert numerical results for Feynman integrals to exact evaluations, when one has reason to suspect the existence of reductions to linear combinations of a basis, with rational or algebraic coefficients. Once a tentative reduction is obtained, confidence in its validity is greatly increased by computing more decimal digits of the terms and verifying the stability of the result. Here we give examples of how the PSLQ and LLL algorithms have yielded remarkable reductions of Feynman integrals to multiple polylogarithms and to the periods and quasi-periods of modula…
We decide to combine this preprint with two other ones (arXiv:1101.1946 and arXiv:1008.3887). The results on Apery numbers and polynomials will be added to the new version of arxiv:1101.1946, and those results on generalized trinomial coefficients will be added to the new version of arxiv:1008.3887
For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Let $p$ be an odd prime. In this paper, we determine the summation $\sum_{k=0}^{p-1}T_k(b,c)^2/m^k$ modulo $p^2$ for integers $m$ with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), 1375--1400].
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