Ji-Cai Liu
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Ji-Cai Liu has a total of 19 co-authors. Here are the top co-authors by number of shared publications:
- Victor J. W. Guo
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, P. R. China
6 shared publications - Chen Wang
Department of Applied Mathematics, Nanjing Forestry University, Nanjing, People’s Republic of China
2 shared publications - Fedor Petrov
St. Petersburg State University, St. Petersburg, Russia
2 shared publications - He-Xia Ni
Department of Applied Mathematics, Nanjing Audit University, Nanjing, People’s Republic of China
2 shared publications - Jing Liu
Department of Mathematics, Wenzhou University, Wenzhou, People’s Republic of China
2 shared publications - Long Li
School of Mathematics and Statistics, Huaiyin Normal University, Huai’an, People’s Republic of China
2 shared publications
Latest Publications
- 9 October 2026
- Published by Springer Nature in Bulletin of the Malaysian Mathematical Sciences Society
- Vol. 49 (5) , 202
Proofs of Two q-Congruence Conjectures of Guo
- 9 October 2026
- Published by Springer Nature in Bulletin of the Malaysian Mathematical Sciences Society
- Vol. 49 (5) , 202
Proofs of Two q-Congruence Conjectures of Guo
- 30 June 2026
- Published by Editura Academiei Române in Proceedings of the Romanian Academy, Series A: Mathematics, Physics, Technical Sciences, Information Science
- Vol. 27 (2) , 123-127
Combinatorial proofs of several partition identities of Andrews and Merca
Based on a bijection due to Fu and Tang, we provide combinatorial proofs of several partition identities of Andrews and Merca.
- 30 June 2026
- Published by Editura Academiei Române in Proceedings of the Romanian Academy, Series A: Mathematics, Physics, Technical Sciences, Information Science
- Vol. 27 (2) , 123-127
Combinatorial proofs of several partition identities of Andrews and Merca
Based on a bijection due to Fu and Tang, we provide combinatorial proofs of several partition identities of Andrews and Merca.
- 16 December 2025
- Published by Akademiai Kiado Zrt. in Studia Scientiarum Mathematicarum Hungarica
- Vol. 62 (4) , 354-360
On 𝓁-Regular Partitions and Hickerson’s Identity
Based on two involutions and a bijection, we combinatorially determine the difference between the number of 𝓁-regular partitions of 𝑛 into an even number of parts and into an odd number of parts for all positive integers 𝑛 and 𝓁 > 1, which extends two recent results due to Ballantine and Merca. As an application, we provide a combinatorial proof of Hickerson’s identity on the number of partitions into an even and odd number of parts.
- 16 December 2025
- Published by Akademiai Kiado Zrt. in Studia Scientiarum Mathematicarum Hungarica
- Vol. 62 (4) , 354-360
On 𝓁-Regular Partitions and Hickerson’s Identity
Based on two involutions and a bijection, we combinatorially determine the difference between the number of 𝓁-regular partitions of 𝑛 into an even number of parts and into an odd number of parts for all positive integers 𝑛 and 𝓁 > 1, which extends two recent results due to Ballantine and Merca. As an application, we provide a combinatorial proof of Hickerson’s identity on the number of partitions into an even and odd number of parts.
- 28 October 2025
- Published by Springer Nature in Journal of Algebraic Combinatorics
- Vol. 62 (3) , 1-24
A general positivity result on coefficients of certain q-series
- 28 October 2025
- Published by Springer Nature in Journal of Algebraic Combinatorics
- Vol. 62 (3) , 1-24
A general positivity result on coefficients of certain q-series
- Open Access
$q$-Binomial expansions of the truncated MacMahon's $q$-series
In 1920, MacMahon introduced two families of $q$-series to study divisor sums. Recent work has shown that MacMahon's $q$-series are closely connected to overpartitions and $3$-colored partitions. Merca introduced truncated forms of MacMahon's $q$-series to generalize earlier results by Andrews-Rose and Ono-Singh, and posed two conjectures regarding the $q$-binomial expansions of these truncated series. In this paper, we provide combinatorial proofs of Merca's conjectures through the combinatorial interpretation of $q$-binomial coefficients.
Open Access$q$-Binomial expansions of the truncated MacMahon's $q$-series
In 1920, MacMahon introduced two families of $q$-series to study divisor sums. Recent work has shown that MacMahon's $q$-series are closely connected to overpartitions and $3$-colored partitions. Merca introduced truncated forms of MacMahon's $q$-series to generalize earlier results by Andrews-Rose and Ono-Singh, and posed two conjectures regarding the $q$-binomial expansions of these truncated series. In this paper, we provide combinatorial proofs of Merca's conjectures through the combinatorial interpretation of $q$-binomial coefficients.
- 28 July 2025
- Published by Cambridge University Press (CUP) in Bulletin of the Australian Mathematical Society
- Vol. 113 (2) , 177-188
ENUMERATION AND DIVISIBILITY OF HIGHER DIMENSIONAL KING WALKS
The lattice walks in the plane starting at the origin $\mathbf {0}$ with steps in $\{-1,0,1\}^{2}\setminus \{\mathbf {0}\}$ are called king walks. We investigate enumeration and divisibility for higher dimensional king walks confined to certain regions. Specifically, we establish an explicit formula for the number of $(r+s)$ -dimensional king walks of length n ending at $(a_1,\ldots ,a_r,b_1,\ldots ,b_s)$ which never dip below $x_i=0$ for $i=1,\ldots ,r$ . We also derive divisibility properties for the number of $(r+s)$ -dimensional king walks of length p (an odd prime) through group actions.- 28 July 2025
- Published by Cambridge University Press (CUP) in Bulletin of the Australian Mathematical Society
- Vol. 113 (2) , 177-188
ENUMERATION AND DIVISIBILITY OF HIGHER DIMENSIONAL KING WALKS
The lattice walks in the plane starting at the origin $\mathbf {0}$ with steps in $\{-1,0,1\}^{2}\setminus \{\mathbf {0}\}$ are called king walks. We investigate enumeration and divisibility for higher dimensional king walks confined to certain regions. Specifically, we establish an explicit formula for the number of $(r+s)$ -dimensional king walks of length n ending at $(a_1,\ldots ,a_r,b_1,\ldots ,b_s)$ which never dip below $x_i=0$ for $i=1,\ldots ,r$ . We also derive divisibility properties for the number of $(r+s)$ -dimensional king walks of length p (an odd prime) through group actions.- 23 June 2025
- Published by Springer Nature in Mediterranean Journal of Mathematics
- Vol. 22 (5) , 1-18
q-Supercongruences Motivated by Carlitz’s Formula
- 23 June 2025
- Published by Springer Nature in Mediterranean Journal of Mathematics
- Vol. 22 (5) , 1-18
q-Supercongruences Motivated by Carlitz’s Formula
- 10 February 2025
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 153 (4) , 1425-1433
A combinatorial approach to Berkovich type identities
Motivated by Berkovich’s nine q q -binomial identities involving the Legendre symbol ( d 3 ) (\frac {d}{3}) , we establish a unified form of q q -binomial identities of this type through a combinatorial approach. This unified form includes Berkovich’s nine identities as special cases. Many new such identities can be also deduced from this unified form.
- 10 February 2025
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 153 (4) , 1425-1433
A combinatorial approach to Berkovich type identities
Motivated by Berkovich’s nine q q -binomial identities involving the Legendre symbol ( d 3 ) (\frac {d}{3}) , we establish a unified form of q q -binomial identities of this type through a combinatorial approach. This unified form includes Berkovich’s nine identities as special cases. Many new such identities can be also deduced from this unified form.
- 8 January 2025
- Published by Canadian Mathematical Society in Canadian Mathematical Bulletin
- Vol. 68 (2) , 359-376
On the binomial transforms of Apéry-like sequences
In his proof of the irrationality of $\zeta (3)$ and $\zeta (2)$ , Apéry defined two integer sequences through $3$ -term recurrences, which are known as the famous Apéry numbers. Zagier, Almkvist–Zudilin, and Cooper successively introduced the other $13$ sporadic sequences through variants of Apéry’s $3$ -term recurrences. All of the $15$ sporadic sequences are called Apéry-like sequences. Motivated by Gessel’s congruences mod $24$ for the Apéry numbers, we investigate congruences of the form $u_n\equiv \alpha ^n \ \pmod {N_{\alpha }}~(\alpha \in \mathbb {Z},N_{\alpha }\in \mathbb {N}^{+})$ for all of the $15$ Apéry-like sequences $\{u_n\}_{n\ge 0}$ . Let $N_{\alpha }$ be the largest positive integer such that $u_n\equiv \alpha ^n\ \pmod {N_{\alpha }}$ for all non-negative integers n. We determine the values of $\max \{N_{\alpha }|\alpha \in \mathbb {Z}\}$ for all of the $15$ Apéry-like sequences $\{u_n\}_{n\ge 0}$ . The binomial transforms of Apéry-like sequences provide us a unified approach to this type of congruences for Apéry-like sequences.- 8 January 2025
- Published by Canadian Mathematical Society in Canadian Mathematical Bulletin
- Vol. 68 (2) , 359-376
On the binomial transforms of Apéry-like sequences
In his proof of the irrationality of $\zeta (3)$ and $\zeta (2)$ , Apéry defined two integer sequences through $3$ -term recurrences, which are known as the famous Apéry numbers. Zagier, Almkvist–Zudilin, and Cooper successively introduced the other $13$ sporadic sequences through variants of Apéry’s $3$ -term recurrences. All of the $15$ sporadic sequences are called Apéry-like sequences. Motivated by Gessel’s congruences mod $24$ for the Apéry numbers, we investigate congruences of the form $u_n\equiv \alpha ^n \ \pmod {N_{\alpha }}~(\alpha \in \mathbb {Z},N_{\alpha }\in \mathbb {N}^{+})$ for all of the $15$ Apéry-like sequences $\{u_n\}_{n\ge 0}$ . Let $N_{\alpha }$ be the largest positive integer such that $u_n\equiv \alpha ^n\ \pmod {N_{\alpha }}$ for all non-negative integers n. We determine the values of $\max \{N_{\alpha }|\alpha \in \mathbb {Z}\}$ for all of the $15$ Apéry-like sequences $\{u_n\}_{n\ge 0}$ . The binomial transforms of Apéry-like sequences provide us a unified approach to this type of congruences for Apéry-like sequences.- Open Access
On the positive coefficients of two families of $q$-series
Let $S$ be a finite set of pairwise coprime positive integers and $Ax^2+Bx$ be an integer valued polynomial with $A> B\ge 0$. For integers $k\ge 1$ and $n\ge 0$, the coefficients $γ_{S,A,B}^k (n)$ are defined as \begin{align*} \prod_{s\in S}\frac{1}{1-q^s}\sum_{j\not\in [-k,k-1]} (-1)^{j+k}q^{Aj^2+Bj}=\sum_{n= 0}^{\infty}γ_{S,A,B}^k (n)q^n. \end{align*} In this paper, we investigate the positivity of $γ_{S,A,B}^k (n)$ for $|S|=4,5$.Open AccessOn the positive coefficients of two families of $q$-series
Let $S$ be a finite set of pairwise coprime positive integers and $Ax^2+Bx$ be an integer valued polynomial with $A> B\ge 0$. For integers $k\ge 1$ and $n\ge 0$, the coefficients $γ_{S,A,B}^k (n)$ are defined as \begin{align*} \prod_{s\in S}\frac{1}{1-q^s}\sum_{j\not\in [-k,k-1]} (-1)^{j+k}q^{Aj^2+Bj}=\sum_{n= 0}^{\infty}γ_{S,A,B}^k (n)q^n. \end{align*} In this paper, we investigate the positivity of $γ_{S,A,B}^k (n)$ for $|S|=4,5$.