arXiv subjects

Julia Q. D. Du

Publications and source records attributed to Julia Q. D. Du.

4 recordsLinked to original sources

arXiv2026

Modular Nahm sums for symmetrizable matrices of indices $({2,\ldots, 2},1)$ and $({1,\ldots, 1},2)$

In this paper, we present three families of modular Nahm sums for symmetrizable matrices with arbitrary rank $r\geq 2$ of indices $({2,\ldots, 2},1)$ and $({1,\ldots, 1},2)$. Specifically, the cases corresponding to $r = 2$ and $r = 3$ of these families have been previously demonstrated by Mizuno, Warnaar, and B. Wang-L. Wang. Building upon these three families, we construct two vector-valued automorphic forms, one of which is a vector-valued modular function when $r$ is odd.

math.CO↗

arXiv2025

Congruences modulo arbitrary powers of $5$ and $7$ for Andrews and Paule's partition diamonds with $(n+1)$ copies of $n$

Recently, Andrews and Paule introduced a partition function $PDN1(N)$ which denotes the number of partition diamonds with $(n+1)$ copies of $n$ where summing the parts at the links gives $N$. They also presented the generating function for $PDN1(n)$ and proved several congruences modulo 5,7,25,49 for $PDN1(n)$. At the end of their paper, Andrews and Paule asked for determining infinite families of congruences similar to Ramanujan's classical $ p(5^kn +d_k) \equiv 0 \pmod {5^k}$, where $24d_k\equiv 1 \pmod {5^k}$ and $k\geq 1$. In this paper, we give an answer of Andrews and Paule's open problem by proving three congruences modulo arbitrary powers of $5$ for $PDN1(n)$. In addition, we prove two congruences modulo arbitrary powers of $7$ for $PDN1(n)$, which are analogous to Watson's congruences for $p(n)$.

math.NT↗

arXiv2019

Finding Modular Functions for Ramanujan-Type Identities

This paper is concerned with a class of partition functions $a(n)$ introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu's algorithms, we present an algorithm to find Ramanujan-type identities for $a(mn+t)$. While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for $p(11n+6)$ with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions $\overline{p}(5n+2)$ and $\overline{p}(5n+3)$ and Andrews--Paule's broken $2$-diamond partition functions $\triangle_{2}(25n+14)$ and $\triangle_{2}(25n+24)$. It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews' singular overpartition functions $\overline{Q}_{3,1}(9n+3)$ and $ \overline{Q}_{3,1}(9n+6)$ due to Shen, the $2$-dissection formulas of Ramanujan and the $8$-dissection formulas due to Hirschhorn.

math.NT↗

arXiv2018

Congruences for the Coefficients of the Powers of the Euler Product

Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we determine the generating functions of $p_{8k}(2^{2α} n +\frac{k(2^{2α}-1)}{3})$ $(1\leq k\leq 3)$ and $p_{3k} (3^{2β}n+\frac{k(3^{2β}-1)}{8})$ $(1\leq k\leq 8)$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo $m$, we obtain infinite families of congruences for $p_k(n)$ modulo any $m\geq2$, where $1\leq k\leq 24$ and $3|k$ or $8|k$. Based on these congruences for $p_k(n)$, infinite families of congruences for many partition functions such as the overpartition function, $t$-core partition functions and $\ell$-regular partition functions are easily obtained.

math.CO↗