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Ji-Cai Liu

Publications and source records attributed to Ji-Cai Liu.

At least 19 recordsLinked to original sources
arXiv2026

Truncated MacMahon-type $q$-series in arithmetic progressions and Chebyshev expansions

We study three finite families of MacMahon-type $q$-series. The first is the arithmetic-progression family $B_{k,m,N,r}^{\pm}(q)$, which contains the usual truncated MacMahon series and their odd-part analogues as special cases. The other two families, $A_{k,m,a}(q)$ and $C_{k,m,a}(q)$, replace the squared linear factors by quadratic factors and are naturally controlled by Chebyshev polynomials. We prove a double $q$-binomial expansion for $B_{k,m,N,r}^{\pm}(q)$ by a weighted elementary-symmetric-function argument; this proves and extends Merca's Conjecture 7 on truncated MacMahon series. We also establish finite Chebyshev polynomial expansions for $A_{k,m,a}(q)$ and $C_{k,m,a}(q)$, and we give weighted partition proofs of the $a=-2$ identities for both the ordinary family $A_{k,m,a}(q)$ and the odd family $C_{k,m,a}(q)$.

math.CO↗
arXiv2026

Two $q$-congruences from Jackson's ${}_8ϕ_7$ summation and Andrews' ${}_4ϕ_3$ summation

We prove two $q$-congruence conjectures of Guo on truncated basic hypergeometric series. The first result strengthens a congruence obtained from Jackson's terminating very-well-poised ${}_8ϕ_7$ summation from the modulus $Φ_n(q)^4$ to the modulus $Φ_n(q)^5$ when $n\equiv3\pmod 5$. The second result proves a cyclotomic congruence modulo $Φ_n(q)$ when $n\equiv1\pmod4$, by a specialization of Andrews' terminating ${}_4ϕ_3$ summation.

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arXiv2026

Proofs of two $q$-congruence conjectures of Guo

We prove two conjectural $q$-congruences proposed by Guo. The first is Conjecture 7.2 in Guo's work on $q$-analogues of two ``divergent'' Ramanujan-type supercongruences; it asserts a square-cyclotomic congruence for a truncated $q$-analogue of a Ramanujan-type sum when $n\equiv1\pmod4$. The second is Conjecture 4.1 in Guo's extension of Van Hamme's $(A.2)$ supercongruence; it gives divisibility modulo $[n]$ for a family of truncated basic hypergeometric sums with a parameter $s$. The proof of the first result relies on a known Watson-transformation congruence obtained by Guo. The proof of the second result is based on period decomposition at primitive roots of unity and a reflection cancellation inside residue blocks.

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arXiv2026

Proofs of Two Positivity Conjectures of Guo

We prove two positivity conjectures proposed by Guo for alternating sums and factorial ratios built from Gaussian coefficients. The first result proves the positivity of the odd $q$-super Catalan numbers \[ C_{m,n}(q)=\frac{[2m+1]![2n]!}{[m+n+1]![m]![n]!}. \] The proof uses the positivity theorem of Warnaar and Zudilin for the usual $q$-super Catalan numbers, together with two recurrences obtained from a double application of the $q$-Chu--Vandermonde summation. The second result proves Guo's conjectural strengthening of his alternating-sum positivity theorem, replacing the exponent coefficient $2r-1$ by every odd coefficient $2b-1$, $1\leq b\leq r$. Its proof combines a $q\mapsto q^{-1}$ reciprocity with a finite deletion recurrence.

math.CO↗
arXiv2024

A general positivity result on coefficients of certain $q$-series

Based on a classical result on partitions of an integer into a finite set of positive integers, we establish a general positivity result on coefficients of certain $q$-series which uniformly refines the positivity of truncated pentagonal number series, truncated Gauss' identities and some special cases of truncated Jacobi triple product identity. As an application, we prove two positivity conjectures due to Merca.

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arXiv2024

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 $|S|=4,5$.

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arXiv2024

On the binomial transforms of Apéry-like sequences

In the proof of the irrationality of $ζ(3)$ and $ζ(2)$, 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 the congruences in the form $u_n\equiv α^n \pmod{N_α}~(α\in \mathbb{Z},N_α\in \mathbb{N}^{+})$ for all of the $15$ Apéry-like sequences $\{u_n\}_{n\ge 0}$. Let $N_α$ be the largest positive integer such that $u_n\equiv α^n \pmod{N_α}$ for all non-negative integers $n$. We determine the values of $\max\{N_α|α\in \mathbb{Z}\}$ 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.

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arXiv2024

A bijection proof of Andrews-Merca integer partition theorem

Andrews and Merca [J. Combin. Theory Ser. A 203 (2024), Art. 105849] recently obtained two interesting results on the sum of the parts with the same parity in the partitions of $n$ (the modulo $2$ case), the proof of which relies on generating functions. Motivated by Andrews and Merca's results, we define six statistics related to the partitions of $n$ and show that the two triples of the six statistics are equidistributed. From this equidistributed result, we derive modulo $m$ extensions of Andrews and Merca's results for all integers $m\ge 2$. The proof of the main result is based on a general bijection on the set of partitions of $n$.

math.CO↗
arXiv2024

On $\ell$-regular partitions and Hickerson's identity

Based on two involutions and a bijection, we completely determine the difference between the number of $\ell$-regular partitions of $n$ into an even number of parts and into an odd number of parts for all positive integers $n$ and $\ell>1$, 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.

math.CO↗
arXiv2024

An extension of Gauss congruences for Apéry numbers

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 Apéry numbers and four kinds of Apéry-like numbers as special cases, and showed that the numbers $\{A_n^{(r,s,t)}\}_{n\ge 0}$ satisfy the Gauss congruences of order $3$. We establish an extension of Osburn--Sahu--Straub congruence through Bernoulli numbers, which is one step deep congruence of the Gauss congruence for $A_n^{(r,s,t)}$.

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arXiv2024

A combinatorial approach to Berkovich type identities

Motivated by Berkovich's nine $q$-binomial identities involving the Legendre symbol $(\frac{d}{3})$, we establish a unified form of $q$-binomial identities of this type through a combinatorial approach. This unified form includes Berkovich's nine identities as special cases. Many such identities can be also deduced from this unified form.

math.CO↗
arXiv2022

Supercongruences involving Motzkin numbers and central trinomial coefficients

Let $M_n$ and $T_n$ denote the $n$th Motzkin number and the $n$th central trinomial coefficient respectively. We prove that for any prime $p\ge 5$, \begin{align*} &\sum_{k=0}^{p-1}M_k^2\equiv \left(\frac{p}{3}\right)\left(2-6p\right)\pmod{p^2},\\ &\sum_{k=0}^{p-1}kM_k^2\equiv \left(\frac{p}{3}\right)\left(9p-1\right)\pmod{p^2},\\ &\sum_{k=0}^{p-1}T_kM_k\equiv \frac{4}{3}\left(\frac{p}{3}\right)+\frac{p}{6}\left(1-9\left(\frac{p}{3}\right)\right)\pmod{p^2}, \end{align*} where $\left(-\right)$ is the Legendre symbol. These results confirm three 12-year-old supercongruence conjectures of Z.-W. Sun.

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arXiv2022

Further results on the divisibility of $q$-trinomial coefficients

We study divisibility for the $q$-trinomial coefficients $τ_0(n,m,q)$, $T_0(n,m,q)$ and $T_1(n,m,q)$, which were first introduced by Andrews and Baxter. In particular, we completely determine $τ_0(an,bn,q)$, $T_0(an,bn,q)$ and $T_1(an,bn,q)$ modulo the square of the cyclotomic polynomial $Φ_n(q)$ for $(a,b)=(m,m-1)$.

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arXiv2021

On the divisibility of $q$-trinomial coefficients

We establish a congruence on sums of central $q$-binomial coefficients. From this $q$-congruence, we derive the divisibility of the $q$-trinomial coefficients introduced by Andrews and Baxter.

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