Related papers: Congruences and relations for r-Fishburn numbers
The Fishburn numbers $\xi (n)$ are defined by the formal power series \[ \sum_{n \geq 0} \xi (n) q^n = \sum_{n \geq 0} \prod_{j = 1}^n (1 - (1 - q)^j). \] Recently, G. Andrews and J. Sellers discovered congruences of the form $\xi (p m + j)…
Recently, a beautiful paper of Andrews and Sellers has established linear congruences for the Fishburn numbers modulo an infinite set of primes. Since then, a number of authors have proven refined results, for example, extending all of…
Andrews and Sellers recently initiated the study of arithmetic properties of Fishburn numbers. In this paper, we prove prime power congruences for generalized Fishburn numbers. These numbers are the coefficients in the $1-q$ expansion of…
There has been significant recent interest in the arithmetic properties of the coefficients of $F(1-q)$ and $\mathscr{F}_t(1-q)$ where $F(q)$ is the Kontsevich-Zagier strange series and $\mathscr{F}_t(q)$ is the strange series associated to…
By using Andrews's explicit formulae of the $q$-Fibonacci sequence introduced by Schur, we prove certain congruences of the $q$-Fibonacci sequence which relate the sequence with the original Fibonacci sequence. As a corollary, we show that…
In 2012 Paule and Radu proved a difficult family of congruences modulo powers of 5 for Andrews' 2-colored generalized Frobenius partition function. The family is associated with the classical modular curve of level 20. We demonstrate the…
We give a new proof of a_4\phi_3 summation due to G.E. Andrews and confirm another_4\phi_3 summation conjectured by him recently. Some variations of these two_4\phi_3 summations are also given.
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…
We present two new Ramanujan-type congruences modulo 5 for overpartition. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition.
In terms of Sear's transformation formula for $_4\phi_3$-series, we give new proofs of a summation formula for ${_4\phi_3}$-series due to Andrews [2] and another summation formula for${_4\phi_3}$-series conjectured in the same paper.…
Let $p_{\{3, 3\}}(n)$ denote the number of $3$-regular partitions in three colours. In a very recent paper, da Silva and Sellers studied certain arithmetic properties of $p_{\{3, 3\}}(n)$. They further conjectured four Ramanujan-like…
Andrews, Brietzke, R\o dseth and Sellers proved an infinite family of congruences on the number of the restricted $m$-ary partitions when $m$ is a prime. In this note, we show that these congruences hold for arbitrary positive integer $m$…
In this paper we present many congruences for several Ap\'ery-like sequences.
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
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 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) =…
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.
Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle…
Osburn, Sahu and Straub introduced the numbers: \begin{align*} A_n^{(r,s,t)}=\sum_{k=0}^n{n\choose k}^r{n+k\choose k}^s{2k\choose n}^t, \end{align*} for non-negative integers $n,r,s,t$ with $r\ge 2$, which includes two kinds of Ap\'ery…
Let $NT(m, k, n)$ denote the total number of parts in the partitions of n with rank congruent to m modulo k. Andrews proved Beck's conjecture on congruences for $NT(m, k, n)$ modulo 5 and 7. Generalizing Andrews'results, Chern obtain…