Andrews-Bressoud series and Wronskians

Andrews-Gordon级数的Wronskian的数论性质

Number theoretic properties of Wronskians of Andrews-Gordon series

对于正整数$1\leq i\leq k$，我们考虑了Wronskians的导数在Andrews-Gordon $q$ -级数$$ \prod_{1\leq n\not \equiv 0,\pm i\pmod{2k+1}}\frac{1}{1-q^n}. $$的某些规范化中的算术性质。这项研究的动机是它们在共形场论中的出现，其中这些级数本质上是$(2,2k+1)$ Virasoro极小模型的不可约特征标。我们确定这样的Wronskians消失，其证明结果揭示了许多分区身份。例如，如果$P_{b}(a;n)$表示$n$划分为不全等$0, \pm a\pmod b$的部分的数目，则对于每个正整数$n$，我们有$$ P_{27}(12; n)=P_{27}(6;n-1) + P_{27}(3;n-2). $$。我们还表明，这些子项将超奇异椭圆曲线分类为特征$p$。更准确地说，如果$2k+1=p$，其中$p\geq 5$是素数，并且商是非零的，那么它本质上是特征$p$超奇异$j$ -不变量在特征$p$中的轨迹。

Bressoud-Subbarao type weighted partition identities for a generalized divisor function

In 1984, Bressoud and Subbarao obtained an interesting weighted partition identity for a generalized divisor function, by means of combinatorial arguments. Recently, the last three named authors found an analytic proof of the aforementioned identity of Bressoud and Subbarao starting from a $q$-series identity of Ramanujan. In the present paper, we revisit the combinatorial arguments of Bressoud and Subbarao, and derive a more general weighted partition identity. Furthermore, with the help of a fractional differential operator, we establish a few more Bressoud-Subbarao type weighted partition i…

On certain identities involving Nahm-type sums with double poles

We prove certain Nahm-type sum representations for the (odd modulus) Andrews-Gordon identities, the (even modulus) Andrews-Bressoud identities, and Rogers' false theta functions. These identities are motivated on one hand by a recent work of C. Jennings-Shaffer and one of us on double pole series, and, on the other hand, by Córdova, Gaiotto and Shao's work on defect Schur's indices.

A finite analogue of a $q$-series identity of Bhoria, Eyyunni and Maji and its applications

Bhoria, Eyyunni and Maji recently obtained a four-parameter $q$-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical proof of a result of Bressoud and Subbarao. Here, we obtain a finite analogue of their identity which naturally gives finite analogues of Ramanujan's results. Using one of these finite analogues, we deduce an identity for a finite sum involving a ${}_2ϕ_1$. This identity is then applied to obtain a generalization of the generating function version of Andrews' f…

Hecke-type series involving infinite products

Since the study by Jacobi and Hecke, Hecke-type series have received extensive attention. Especially, Hecke-type series involving infinite products have attracted broad interest among many mathematicians including Kac, Peterson, Andrews, Bressoud and Liu. Motivated by the works of these people, we study Hecke-type series involving infinite products. In particular, we establish some Hecke-type series involving infinite products and then obtain the truncated versions of these series as well as some other known series of the same type. As consequences, three families of inequalities for certain p…

柱分区的MacMahon分析视角

A MacMahon Analysis View of Cylindric Partitions

本文利用MacMahon分区分析研究双元素轮廓柱分区，求得其生成函数显式公式，推广了相关恒等式，并猜想Andrews-Gordon与Bressoud恒等式的新多项式加细，同时利用Bailey引理变体给出新的多项式恒等式无限层级。

作为向量值函数的某些 Nahm 和的模性

Modularity of Some Nahm Sums as Vector-valued Functions

本文通过将若干族 Nahm 和视为向量值函数建立其模变换公式，证明它们在 $N=1,2,3,4$ 的同余子群 $Γ_0(N)$ 上具有模性，并证明了相关变换公式及 theta 级数的向量值变换公式。

Some Observations on Lambert series, vanishing coefficients and dissections of infinite products and series

Andrews and Bressoud, Alladi and Gordon, and others, have proven, in a number of papers, that the coefficients in various arithmetic progressions in the series expansions of certain infinite $q$-products vanish. In the present paper it is shown that these results follow automatically (simply by specializing parameters) in an identity derived from a special case of Ramanujan's $_1ψ_1$ identity. Likewise, a number of authors have proven results about the $m$-dissections of certain infinite $q$-products using various methods. It is shown that many of these $m$-dissections also follow automaticall…