Related papers: Supercongruences for Apery-like numbers
A simplification of Ap\'ery's proof of the irrationality of \zeta(3) is presented. The construction of approximations is motivated from the viewpoint of 2-dimensional recurrence relations which simplifies many of the details of the proof.…
An important component of Ap\'ery's proof that $\zeta (3)$ is irrational involves representing $\zeta (3)$ as the limit of the quotient of two rational solutions to a three-term recurrence. We present various approaches to such Ap\'ery…
Many authors have investigated the congruence relations amongst the coefficients of power series expansions of modular forms $f$ in modular functions $t$. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and…
In this paper, we prove two conjectural supercongruences on the $(p-1)$th Ap\'ery number, which were recently proposed by Z.-H. Sun.
In this paper, we formally introduce the notion of Ap{\'e}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…
In this article, we introduce congruential Euler numbers, which are a further generalization of generalized Euler numbers. We prove the $p$-adic congruences of congruential Euler numbers, which include answers to a conjecture related to…
Ap\'ery's remarkable discovery of rapidly converging continued fractions with small coefficients for $\zeta(2)$ and $\zeta(3)$ has led to a flurry of important activity in an incredible variety of different directions. Our purpose is to…
We derive an expression for the value $\zeta_Q(3)$ of the spectral zeta function $\zeta_Q(s)$ studied by Ichinose and Wakayama for the non-commutative harmonic oscillator defined in the work of Parmeggiani and Wakayama using a Gaussian…
In 1982, Gessel showed that the Ap\'ery numbers associated to the irrationality of $\zeta(3)$ satisfy Lucas congruences. Our main result is to prove corresponding congruences for all sporadic Ap\'ery-like sequences. In several cases, we are…
Using WZ forms, Apery-style proofs of the irrationality of the q-analogues of the Harmonic seires and Ln(2) are given. For the q-analogue of Ln(2), this method of proof produces an improved irrationality measure.
In this paper, we study some Euler-Ap\'ery-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…
We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table of…
This monograph is intended to be considered as my habilitation (D.Sc.) thesis; because of that and as everything has already appeared in English, it is performed exclusively in Russian. The monograph comprises a detailed introduction and…
We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Ap\'{e}ry numbers, by means of a general…
Recently, using modular forms F. Beukers posed a unified method that can deal with a large number of supercongruences involving binomial coefficients and Ap\'ery-like numbers. In this paper, we use Beukers' method to prove some conjectures…
In this paper we present many congruences for several Ap\'ery-like sequences.
A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $\zeta(2)$ and $\zeta(3)$, as well as to explain…
This survey text deals with irrationality, and linear independence over the rationals, of values at positive odd integers of Riemann zeta function. The first section gives all known proofs (and connections between them) of Ap\'ery's Theorem…
We present a new `elementary' proof of the irrationality of $\zeta(3)$ based on some recent `hypergeometric' ideas of Yu.Nesterenko, T.Rivoal, and K.Ball, and on Zeilberger's algorithm of creative telescoping.
The idea to use classical hypergeometric series and, in particular, well-poised hypergeometric series in diophantine problems of the values of the polylogarithms has led to several novelties in number theory and neighbouring areas of…