arXiv subjects

Liuquan Wang

Publications and source records attributed to Liuquan Wang.

At least 37 records · Page 2Linked to original sources

arXiv2023

Proofs of Mizuno's Conjectures on Generalized Rank Two Nahm Sums

Recently, Mizuno studied generalized Nahm sums associated with symmetrizable matrices. He provided 14 sets of candidates of modular Nahm sums in rank two and justified four of them. We prove the modularity for eight other sets of candidates and present conjectural formulas for the remaining two sets of candidates. This is achieved by finding Rogers-Ramanujan type identities associated with these Nahm sums. We also prove Mizuno's conjectural modular transformation formula for a vector-valued function consists of Nahm sums. Meanwhile, we find some new non-modular identities for some other Nahm sums associated with the matrices in Mizuno's candidates.

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arXiv2023

Rogers-Ramanujan type identities involving double, triple and quadruple sums

We prove a number of new Rogers-Ramanujan type identities involving double, triple and quadruple sums. They were discovered after an extensive search using Maple. The main idea of proofs is to reduce them to some known identities in the literature. This is achieved by direct summation or the constant term method. We also obtain some new single-sum identities as consequences.

math.CO↗

arXiv2023

On Some Double Nahm Sums of Zagier

Zagier provided eleven conjectural rank two examples for Nahm's problem. All of them have been proved in the literature except for the fifth example, and there is no $q$-series proof for the tenth example. We prove that the fifth and the tenth examples are in fact equivalent. Then we give a $q$-series proof for the fifth example, which confirms a recent conjecture of Wang. This also serves as the first $q$-series proof for the tenth example, whose explicit form was conjectured by Vlasenko and Zwegers in 2011.

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arXiv2023

Multi-sum Rogers-Ramanujan Type Identities

We use an integral method to establish a number of Rogers-Ramanujan type identities involving double and triple sums. The key step for proving such identities is to find some infinite products whose integrals over suitable contours are still infinite products. The method used here is motivated by Rosengren's proof of the Kanade-Russell identities.

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arXiv2023

Modularity of Nahm Sums for the Tadpole Diagram

We prove Rogers-Ramanujan type identities for the Nahm sums associated with the tadpole Cartan matrix of rank $3$. These identities reveal the modularity of these sums, and thereby we confirm a conjecture of Penn, Calinescu and the first author in this case. We show that these Nahm sums together with some shifted sums can be combined into a vector-valued modular function on the full modular group. We also present some conjectures for a general rank.

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arXiv2022

Explicit Forms and Proofs of Zagier's Rank Three Examples for Nahm's Problem

Let $r\geq 1$ be a positive integer, $A$ a real positive semi-definite symmetric $r\times r$ rational matrix, $B$ a rational vector of length $r$, and $C$ a rational scalar. Nahm's problem is to find all triples $(A,B,C)$ such that the $r$-fold $q$-hypergeometric series $$f_{A,B,C}(q):=\sum_{n=(n_1,\dots,n_r)^\mathrm{T}\in (\mathbb{Z}_{\geq 0})^r} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}}$$ becomes a modular form, and we call such $(A,B,C)$ a modular triple. When the rank $r=3$, after extensive computer searches, Zagier provided twelve sets of conjectural modular triples and proved three of them. We prove a number of Rogers-Ramanujan type identities involving triple sums. These identities give modular form representations for and thereby verify all of Zagier's rank three examples. In particular, we prove a conjectural identity of Zagier.

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arXiv2022

New Proofs of Some Double Sum Rogers-Ramanujan Type Identities

Recently, Rosengren utilized an integral method to prove a number of conjectural identities found by Kanade and Russell. Using this integral method, we give new proofs to some double sum identities of Rogers-Ramanujan type. These identities were earlier proved by approaches such as combinatorial arguments or by using $q$-difference equations. Our proofs are based on streamlined calculations, which relate these double sum identities to some known Rogers-Ramanujan type identities with single sums. Moreover, we prove a conjectural identity of Andrews and Uncu which was earlier confirmed by Chern.

math.CO↗

arXiv2021

Sign Changes of Coefficients of Powers of the Infinite Borwein Product

We denote by $c_t^{(m)}(n)$ the coefficient of $q^n$ in the series expansion of $(q;q)_\infty^m(q^t;q^t)_\infty^{-m}$, which is the $m$-th power of the infinite Borwein product. Let $t$ and $m$ be positive integers with $m(t-1)\leq 24$. We provide asymptotic formula for $c_t^{(m)}(n)$, and give characterizations of $n$ for which $c_t^{(m)}(n)$ is positive, negative or zero. We show that $c_t^{(m)}(n)$ is ultimately periodic in sign and conjecture that this is still true for other positive integer values of $t$ and $m$. Furthermore, we confirm this conjecture in the cases $(t,m)=(2,m),(p,1),(p,3)$ for arbitrary positive integer $m$ and prime $p$.

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arXiv2021

Modular Proofs of Gosper's Identities

We give unified modular proofs to all of Gosper's identities on the $q$-constant $Π_q$. We also confirm Gosper's observation that for any distinct positive integers $n_1,\cdots,n_m$ with $m\geq 3$, $Π_{q^{n_1}}$, $\cdots$, $Π_{q^{n_m}}$ satisfy a nonzero homogeneous polynomial. Our proofs provide a method to rediscover Gosper's identities. Meanwhile, several results on $Π_q$ found by El Bachraoui have been corrected. Furthermore, we illustrate a strategy to construct some of Gosper's identities using hauptmoduls for genus zero congruence subgroups.

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arXiv2021

Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

math.CO↗

arXiv2021

Parity of coefficients of mock theta functions

We study the parity of coefficients of classical mock theta functions. Suppose $g$ is a formal power series with integer coefficients, and let $c(g;n)$ be the coefficient of $q^n$ in its series expansion. We say that $g$ is of parity type $(a,1-a)$ if $c(g;n)$ takes even values with probability $a$ for $n\geq 0$. We show that among the 44 classical mock theta functions, 21 of them are of parity type $(1,0)$. We further conjecture that 19 mock theta functions are of parity type $(\frac{1}{2},\frac{1}{2})$ and 4 functions are of parity type $(\frac{3}{4},\frac{1}{4})$. We also give characterizations of $n$ such that $c(g;n)$ is odd for the mock theta functions of parity type $(1,0)$.

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arXiv2021

Modular Forms and $k$-colored Generalized Frobenius Partitions

Let $k$ and $n$ be positive integers. Let $cϕ_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$ and $\mathrm{C}Φ_k(q)$ be the generating function of $cϕ_{k}(n)$. In this article, we study $\mathrm{C}Φ_k(q)$ using the theory of modular forms and discover new surprising properties of $\mathrm{C}Φ_k(q)$.

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arXiv2021

Brouwer degree for Kazdan-Warner equations on a connected finite graph

We study Kazdan-Warner equations on a connected finite graph via the method of the degree theory. Firstly, we prove that all solutions to the Kazdan-Warner equation with nonzero prescribed function are uniformly bounded and the Brouwer degree is well defined. Secondly, we compute the Brouwer degree case by case. As consequences, we give new proofs of some known existence results for the Kazdan-Warner equation on a connected finite graph.

math.DG↗

arXiv2021

A Family of Supercongruences Involving Multiple Harmonic Sums

In recent years, the congruence $$ \sum_{\substack{i+j+k=p\\ i,j,k>0}} \frac1{ijk} \equiv -2 B_{p-3} \pmod{p}, $$ first discovered by the last author have been generalized by either increasing the number of indices and considering the corresponding supercongruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar supercongruences modulo prime powers $p^r$ with the indexes summing up to $mp^r$ where $m$ is coprime to $p$, where all the indexes are also coprime to $p$.

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arXiv2020

Farey sequence and Graham's conjectures

Let ${F}_{n}$ be the Farey sequence of order $n$. For $S \subseteq {F}_n$ we let $\mathcal{Q}(S) = \left\{x/y:x,y\in S, x\le y \, \, \textrm{and} \, \, y\neq 0\right\}$. We show that if $\mathcal{Q}(S)\subseteq F_n$, then $|S|\leq n+1$. Moreover, we prove that in any of the following cases: (1) $\mathcal{Q}(S)=F_n$; (2) $\mathcal{Q}(S)\subseteq F_n$ and $|S|=n+1$, we must have $S = \left\{0,1,\frac{1}{2},\ldots,\frac{1}{n}\right\}$ or $S=\left\{0,1,\frac{1}{n},\ldots,\frac{n-1}{n}\right\}$ except for $n=4$, where we have an additional set $\{0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}\}$ for the second case. Our results are based on Graham's GCD conjectures, which have been proved by Balasubramanian and Soundararajan.

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arXiv2020

Positivity and divisibility of alternating descent polynomials

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also found a quadratic recursion for the alternating major index $q$-analog of the alternating descent polynomials. As an interesting application of this quadratic recursion, we show that $(1+q)^{\lfloor n/2\rfloor}$ divides $\sum_{π\in\mathfrak{S}_n}q^{\rm{altmaj}(π)}$, where $\mathfrak{S}_n$ is the set of all permutations of $\{1,2,\ldots,n\}$ and $\rm{altmaj}(π)$ is the alternating major index of $π$. This leads us to discover a $q$-analog of $n!=2^{\ell}m$, $m$ odd, using the statistic of alternating major index. Moreover, we study the $γ$-vectors of the alternating descent polynomials by using these two recursions and the ${\textbf{cd}}$-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.

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arXiv2020

Modular units and cuspidal divisor classes on $X_0(n^2M)$ with $n|24$ and $M$ squarefree

For a positive integer $N$, let $\mathscr C(N)$ be the subgroup of $J_0(N)$ generated by the equivalence classes of cuspidal divisors of degree $0$ and $\mathscr C(N)(\mathbb Q):=\mathscr C(N)\cap J_0(N)(\mathbb Q)$ be its $\mathbb Q$-rational subgroup. Let also $\mathscr C_{\mathbb Q}(N)$ be the subgroup of $\mathscr C(N)(\mathbb Q)$ generated by $\mathbb Q$-rational cuspidal divisors. We prove that when $N=n^2M$ for some integer $n$ dividing $24$ and some squarefree integer $M$, the two groups $\mathscr C(N)(\mathbb Q)$ and $\mathscr C_{\mathbb Q}(N)$ are equal. To achieve this, we show that all modular units on $X_0(N)$ on such $N$ are products of functions of the form $η(mτ+k/h)$, $mh^2|N$ and $k\in\mathbb Z$ and determine the necessary and sufficient conditions for products of such functions to be modular units on $X_0(N)$.

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arXiv2020

Ramanujan-type $1/π$-series from bimodular forms

We develop an approach to establish $1/π$-series from bimodular forms. Utilizing this approach, we obtain new families of $2$-variable $1/π$-series associated to Zagier's sporadic Apery-like sequences.

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