Related papers: Supercongruences for Apery-like numbers

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) =…

In 1978, Apery has given sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial…

One of the many remarkable properties of the Ap\'ery numbers $A (n)$, introduced in Ap\'ery's proof of the irrationality of $\zeta (3)$, is that they satisfy the two-term supercongruences \begin{equation*} A (p^r m) \equiv A (p^{r - 1} m)…

A generalization of the Apery-like numbers, which is used to describe the special values $\zeta_Q(2)$ and $\zeta_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a…

Using techniques due to Coster, we prove a supercongruence for a generalization of the Domb numbers. This extends a recent result of Chan, Cooper and Sica and confirms a conjectural supercongruence for numbers which are coefficients in one…

It is well-known that the Ap\'ery sequences which arise in the irrationality proofs for $\zeta(2)$ and $\zeta(3)$ satisfy many intriguing arithmetic properties and are related to the $p$th Fourier coefficients of modular forms. In this…

Recently, R. Tauraso established finite $p$-analogues of famous Ap\'ery series for $\zeta(2)$ and $\zeta(3).$ In this paper, we present several congruences for finite central binomial sums arising from the truncation of Ap\'ery-type series…

We establish supercongruences for two kinds of Ap\'ery-like numbers, which involve Bernoulli numbers and Bernoulli polynomials. Conjectural supercongruences of the same type for another four kinds of Ap\'ery-like numbers are also proposed.

In this article we shall survey some recent progress on the study of Ap\'ery-like sums which are multiple variable generalizations of the two sums Ap\'ery used in his famous proof of the irrationality of $\zeta(2)$ and $\zeta(3)$. We only…

In the proof of the irrationality of $\zeta(3)$ and $\zeta(2)$, Ap\'ery defined two integer sequences through $3$-term recurrences, which are known as the famous Ap\'ery numbers. Zagier, Almkvist--Zudilin and Cooper successively introduced…

Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for zeta(4n+3) which generalizes Apery's series for zeta(3), and appears to give the best possible series…

In this note we evaluate multiple integrals that play a crucial role in the theory of irrationality of zeta function

New (infinitely many) rational approximants to \zeta(3) proving its irrationality are given. The recurrence relations for the numerator and denominator of these approximants as well as their continued fraction expansions are obtained. A…

Roger Apery's seminal method for proving irrationality is "turned on its head" and taught to computers, enabling a one second redux of the original proof of zeta(3), and many new irrationality proofs of many new constants, alas, none of…

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…

In this note, I develop step-by-step proofs of irrationality for $\,\zeta{(2)}\,$ and $\,\zeta{(3)}$. Though the proofs follow closely those based upon unit-square integrals proposed originally by Beukers, I introduce some modifications…

By using the Wilf-Zeilberger method, we prove a novel finite combinatorial identity related to a bivariate generating function for $\zeta(2+r+2s)$ (an extension of a Bailey-Borwein-Bradley Apery-like formula for even zeta values). Such…

In the 1980s, Koecher and, independently, Leshchiner found an elegant formula for the generating function of odd zeta values. In this short note, we derive a $q$-analogue of this formula, which provides a $q$-version of the accelerated…

For each of the $15$ known sporadic Ap\'ery-like sequences, we prove congruences modulo $p^2$ that are natural extensions of the Lucas congruences modulo $p$. This extends a result of Gessel for the numbers used by Ap\'ery in his proof of…

Many generating series of combinatorially interesting numbers have the property that the sum of the terms of order $<p$ at some suitable point is congruent to a zero of a zeta-function modulo infinitely many primes $p$. Surprisingly, very…