6 papers · 1 filter
Two -congruences from Jackson's summation and Andrews' summation
Ji-Cai Liu, Qing-Yuan Tao
We prove two -congruence conjectures of Guo on truncated basic hypergeometric series. The first result strengthens a congruence obtained from Jackson's terminating very-well-poi…
Proofs of two -congruence conjectures of Guo
Ji-Cai Liu, Qing-Yuan Tao
We prove two conjectural -congruences proposed by Guo. The first is Conjecture 7.2 in Guo's work on -analogues of two ``divergent'' Ramanujan-type supercongruences; it assert…
On the positive coefficients of two families of -series
Ji-Cai Liu, Kong-Lian Liao
Let be a finite set of pairwise coprime positive integers and be an integer valued polynomial with . For integers and , the coefficients $…
On the binomial transforms of Apéry-like sequences
Ji-Cai Liu
In the proof of the irrationality of and , Apéry defined two integer sequences through -term recurrences, which are known as the famous Apéry numbers. Zagier, Almkv…
An extension of Gauss congruences for Apéry numbers
Ji-Cai Liu
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,…
Supercongruences involving Apéry-like numbers and Bernoulli numbers
Ji-Cai Liu
We establish supercongruences for two kinds of Apéry-like numbers, which involve Bernoulli numbers and Bernoulli polynomials. Conjectural supercongruences of the same type for anot…