Related papers: Sequences, modular forms and cellular integrals
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…
It is known that the numbers which occur in Apery's proof of the irrationality of zeta(2) have many interesting congruence properties while the associated generating function satisfies a second order differential equation. We prove…
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) =…
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…
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…
Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for $\zeta(3)$ in terms of a critical $L$-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of…
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)…
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…
We prove a supercongruence modulo $p^3$ between the $p$th Fourier coefficient of a weight 6 modular form and a truncated ${}_6F_5$-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to…
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…
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…
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…
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…
Following Apery's proof of the irrationality of zeta(3), Beukers found an elegant reinterpretation of Apery's arguments using modular forms. We show how Beukers arguments can be adapted to a p-adic setting. In this context, certain…
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…
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…
In this paper, we will apply the ideas from the mirror symmetry of Calabi-Yau threefolds to study the modular forms and one-parameter family of K3 surfaces found by Beukers and Peters, which provide enlightenment to the two mysterious…
We analyse a certain family of cellular integrals, which are period integrals on the moduli space $\mathcal{M}_{0,8}$ of curves of genus zero with eight marked points, and give rise to simultaneous rational approximations to $\zeta(3)$ and…
A thorough analysis is made of the Fourier coefficients for vector-valued modular forms associated to three-dimensional irreducible representations of the modular group. In particular, the following statement is verified for all but a…
There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work arXiv:1803.01830, such partial sums are related to the radial asymptotics of infinite…