105 citations · 168 across the 35 of their papers we have counts for
32 papers · 1 filter
Two new -supercongruences arising from Carlitz's identity
Ji-Cai Liu, Wei-Wei Qi
From Carlitz's identity, we deduce two new -supercongruences modulo the square of a cyclotomic polynomial, which were originally conjectured by Guo. These results establish new…
Supercongruences involving Motzkin numbers and central trinomial coefficients
Ji-Cai Liu
Let and denote the th Motzkin number and the th central trinomial coefficient respectively. We prove that for any prime , \begin{align*} &\sum_{k=0}^{p-1}…
Further results on the divisibility of -trinomial coefficients
Ji-Cai Liu, Wei-Wei Qi
We study divisibility for the -trinomial coefficients , and , which were first introduced by Andrews and Baxter. In particular, we completel…
On the divisibility of sums of even powers of -binomial coefficients
Ji-Cai Liu, Xue-Ting Jiang
We prove the divisibility conjecture on sums of even powers of -binomial coefficients, which was recently proposed by Guo, Schlosser and Zudilin. Our proof relies on two -har…
An extension of a supercongruence of Long and Ramakrishna
Victor J. W. Guo, Ji-Cai Liu, Michael J. Schlosser
We prove two supercongruences for specific truncated hypergeometric series. These include an uniparametric extension of a supercongruence that was recently established by Long and…
A -adic analogue of Chan and Verrill's formula for
Ji-Cai Liu
We prove three supercongruences for sums of Almkvist-Zudilin numbers, which confirm some conjectures of Zudilin and Z.-H. Sun. A typical example is the Ramanujan-type supercongruen…
Supercongruences for sums involving Domb numbers
Ji-Cai Liu
We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transform…
Supercongruences for Almkvist--Zudilin sequences
Ji-Cai Liu, He-Xia Ni
In this note, we prove two supercongruences involving Almkvist--Zudilin sequences, which were originally conjectured by Z.-H. Sun.
A variation of -Wolstenholme's theorem
Ji-Cai Liu
We investigate a variation of -Wolstenholme's theorem, which extends the -analogue of Wolstenholme's theorem due to Shi and Pan [Amer. Math. Monthly 114 (2007), 529--531]. Th…
On two congruences involving Franel numbers
Ji-Cai Liu
Via symbolic summation method, we establish the following series for : \begin{align*} \sum_{k=1}^\infty \frac{H_k-2H_{2k}}{(-3)^k k} = \frac{π^2}{18}, \end{align*} where $H_k=…
Proof of Sun's supercongruence involving Catalan numbers
Ji-Cai Liu
We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric s…
Some supercongruences arising from symbolic summation
Ji-Cai Liu
Based on some combinatorial identities arising from symbolic summation, we extend two supercongruences on partial sums of hypergeometric series, which were originally conjectured b…
On two supercongruences of double binomial sums
Long Li, Ji-Cai Liu
In this note, we confirm two conjectural supercongruences on double sums of binomial coefficients due to El Bachraoui.
Congruence properties for the trinomial coefficients
Moa Apagodu, Ji-Cai Liu
In this paper, we state and prove some congruence properties for the trinomial coeficients, one of which is similar to the Wolstenholme's theorem.
On a congruence involving -Catalan numbers
Ji-Cai Liu
Based on a -congruence of the author and Petrov, we set up a -analogue of Sun--Tauraso's congruence for sums of Catalan numbers, which extends a -congruence due to Tauraso…
Congruences on sums of -binomial coefficients
Ji-Cai Liu, Fedor Petrov
We establish a -analogue of Sun--Zhao's congruence on harmonic sums. Based on this -congruence and a -series identity, we prove a congruence conjecture on sums of central…
Supercongruences arising from hypergeometric series identities
Ji-Cai Liu
By using some hypergeometric series identities, we prove two supercongruences on truncated hypergeometric series, one of which is related to a modular Calabi--Yau threefold, and th…
On van Hamme's (A.2) and (H.2) supercongruences
Ji-Cai Liu
In 1997, van Hamme conjectured 13 Ramanujan-type supercongruences labeled (A.2)--(M.2). Using some combinatorial identities discovered by Sigma, we extend (A.2) and (H.2) to superc…
Congruences on sums of super Catalan numbers
Ji-Cai Liu
In this paper, we prove two congruences on the double sums of the super Catalan numbers (named by Gessel), which were recently conjectured by Apagodu.
Supercongruences for the th Apéry number
Ji-Cai Liu, Chen Wang
In this paper, we prove two conjectural supercongruences on the th Apéry number, which were recently proposed by Z.-H. Sun.
-Analogues of two Ramanujan-type formulas for
Victor J. W. Guo, Ji-Cai Liu
We give -analogues of the following two Ramanujan-type formulas for : \begin{align*} \sum_{k=0}^\infty (6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 4^k} =\frac{4}π \quad\text{and}\q…
Supercongruences between truncated hypergeometric series
Ji-Cai Liu
We establish four supercongruences between truncated hypergeometric series involving -adic Gamma functions, which extend some of the Rodriguez-Villegas supercongruence…
A -adic supercongruence for truncated hypergeometric series
Ji-Cai Liu
Using an identity due to Gessel and Stanton and some properties of the -adic Gamma function, we establish a -adic supercongruence for truncated hypergeometric series ${}_7F_6…
Supercongruences involving -adic Gamma functions
Ji-Cai Liu
We establish some supercongruences for the truncated and hypergeometric series involving the -adic Gamma functions. Some of these results extend the four Rod…