A generalization of the Apery-like numbers, which is used to describe the special values $ζ_Q(2)$ and $ζ_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a recurrence relation for them, which shows a ladder structure among them. Further, we consider the `rational part' of the higher Apery-like numbers. We discuss several kinds of congruence relations among them, which are regarded as an analogue of the ones among Apery numbers.
In this paper, we study some Euler-Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma functions, polygamma functions and trigonometric functions. The evaluations of a large number of special Euler-Apéry-type series are presented as …
Koecher in 1980 derived a method for obtaining identities for the Riemann zeta function at odd positive integers, including a classical result for $ζ(3)$ due to Markov and rediscovered by Apéry. In this paper we extend Koecher's method to a very general setting and prove two more specific but still rather general results. As applications we obtain infinite classes of identities for alternating Euler sums, further Markov-Apéry type identities, and identities for even powers of $π$
In this paper, we formally introduce the notion of Ap{é}ry-like sums and we show that every multiple zeta values can be expressed as a $\bf Z$-linear combination of them. We even describe a canonical way to do so. This allows us to put in a new theoretical context several identities scattered in the literature, as well as to discover many new interesting ones. We give in this paper new integral formulas for multiple zeta values and Apéry-like sums. They enable us to give a short direct proof of Zagier's formulas for $ζ(2,\ldots,2,3,2,\ldots,2)$ as well as of similar ones in the context of Ap{é…
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 $\…
There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or Zeta(5). Inspired by Frits Beukers' elegant rendition of Apery's seminal proofs of the irrationality of Zeta(2) and Zeta(3), and heavily using algorithmic proof theory, we systematically searched fo…
By using the Wilf-Zeilberger method, we prove a novel finite combinatorial identity related to a bivariate generating function for $ζ(2+r+2s)$ (an extension of a Bailey-Borwein-Bradley Apery-like formula for even zeta values). Such identity is then applied to show several supercongruences.
The purpose of the present paper is to study the number theoretic properties of the special values of the spectral zeta functions of the non-commutative harmonic oscillator (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations, and deepen the understanding of the spectrum of the NcHO. We study first the general expression of special values of the spectral zeta function $ζ_Q(s)$ of the NcHO at $s=n$ $(n=2,3,\dots)$ and then the generating and meta-generating functions for Apéry-like numbers defined through the analysis of speci…
In this paper we present some new identities for multiple polylogarithms (abbr. MPLs) and multiple harmonic star sums (abbr. MHSSs) by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish some explicit relations between Kaneko-Yamamoto type multiple zeta values (abbr. K-Y MZVs), multiple zeta values (abbr. MZVs) and MPLs. Further, we find some explicit relations between MZVs and multiple zeta star values (abbr. MZSVs). Furthermore, we define an Apéry-type variant of MZSVs $ζ^\star_B({\bf k})$ (called multiple zeta $…
In the course of the proof of the irrationality of zeta(2) R. Apery introduced numbers b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}. Stienstra and Beukers showed that for the prime p > 3 Apery numbers satisfy congruence b((p-1)/2) = 4a^2-2p mod p, if p = a^2+b^2 (where a is odd). Later, Zagier found some generalizations of Apery numbers, so called sporadic sequences, and recently Osburn and Straub proved similar congruences for all but one of the six Zagier's sporadic sequences (three cases were already known to be true) and conjectured the congruence for the sixth sequence. In this paper…
We prove a conjecture due to Kimoto and Wakayama from 2006 concerning Apery-like numbers associated to a special value of a spectral zeta function. Our proof uses hypergeometric series and p-adic analysis.
Multiple Clausen values and deformed Apéry-like series
School of Mathematics, Nanjing University（南京大学数学系）；Program in Applied and Computational Mathematics (PACM), Princeton University（普林斯顿大学应用与计算数学项目）；Academy of Advanced Interdisciplinary Studies (AAIS), Peking University（北京大学前沿交叉学科研究院）
We build, using the notion of zinbiel algebra, some commutative subalgebras $C_{u,v}$ inside an algebra of formal iterated integrals. There is a quotient map from this algebra of formal iterated integrals to the algebra of motivic multiple zeta values. Restricting this quotient map to the subalgebras $C_{u,v}$ of A$_{1,0}$ gives a morphism of graded commutative algebras with the same graded dimension. This is conjectured to be generically an isomorphism. When u+v = 0, the image is instead a sub-algebra of the algebra of motivic multiple zeta values.
We study trivial multiple zeta values in Tate algebras. These are particular examples of the multiple zeta values in Tate algebras in positive characteristic introduced by the second author. If the number of variables involved is 'not large' in a way that is made precise in the paper, we can endow the set of trivial multiple zeta values with a structure of module over a non-commutative polynomial ring with coefficients in the rational fraction field over a finite field. We determine the structure of this module in terms of generators and we show how in many cases, this is sufficient for the de…
We study positive characteristic multiple zeta values associated to general curves over $\mathbb F_q$ together with an $\mathbb F_q$-rational point $\infty$ as introduced by Thakur. For the case of the projective line these values were defined as analogues of classical multiple zeta values. In the present paper we first establish a general non-commutative factorization of exponential series associated to certain lattices of rank one. Next we introduce universal families of multiple zeta values of Thakur and show that they are Eulerian in full generality. In particular, we prove a conjecture of…
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 lev…
In the 1980's, Greene defined {\it hypergeometric functions over finite fields} using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the $_2F_1$ functions, t…
设f是一个G-函数（在Siegel意义下），x是一个代数数;假设f（x）是一个实数。作为一个更一般结果的特例，我们证明了f（x）可以写成g（1），其中g是具有任意大收敛半径的有理系数G-函数.作为应用，我们证明了这类值的乘积恰好是序列a（n）/b（n）的极限数，其中这两个序列的生成列都是有理系数的G-函数。这一结果为zeta（3）的无理性证明提供了一个一般的设置，并回答了T. Rivoal in [Approximations rationnelles des valeurs de la fongly Gamma aux rationnels：le cas des puissances，Acta Arith. 142（2010），no. 4，347-365]。
A large family of relations among multiple zeta values may be described using the combinatorics of shuffle and quasi-shuffle algebras. While the structure of shuffle algebras have been well understood for some time now, quasi-shuffle algebras were only formally studied relatively recently. In particular, Hoffman gives a thorough discussion of the algebraic structure, including a choice of algebra basis, and applies his results to produce families of relations among multiple zeta values and their generalisations. In a recent preprint, Hirose and Sato establish a family of relations coming from …
This is a review on the two first parts of our work on $p$-adic multiple zeta values at $N$-th roots of unity ($p$MZV$μ_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,μ_{N},\infty\}$ (where $N$ and $p$ are coprime). We restrict for simplicity the review to the case of $N=1$, i.e. the case of $p$-adic multiple zeta values ($p$MZV's). The main tools are new objects which we call $p$-adic pro-unipotent harmonic actions. These are continuous group actions on a space containing the non-commutative generating series of weighted multiple h…
