MacMahon级数变体的拟模性与极限行为
Quasimodularity and Limiting Behavior for Variations of MacMahon Series
AI总结：
本文利用准洗牌代数理论研究了MacMahon级数变体的扩展族，证明了其拟模性质，确定了精确权重与级别，揭示了其逼近分拆函数的极限行为，并给出了无穷乘积倒数的闭式公式。
摘要：
受MacMahon少校1920年代开创性工作的启发，Amdeberhan--Andrews--Tauraso近期引入了无穷族q-级数$\mathcal{U}_{t}(a;q)$，并证明这些函数是拟模形式的线性组合。本文研究了一个包含该族$\{\mathcal{U}_t\}$的更广q-级数族。利用准洗牌代数理论，我们证明此扩展族也属于拟模形式代数。此外，我们确定了这些函数的精确权重与级别，使前述结果更加锐利。我们进一步探究了这些函数的极限行为，特别地，证明了拟模形式序列$\{\mathcal{U}_t(1;q)\}$可逼近普通分拆函数。我们还建立了用$\mathcal{U}_{t}(a;q)$表示特定无穷乘积倒数的无穷多闭式公式。
英文摘要：
Motivated by the 1920's seminal work of Major MacMahon, Amdeberhan--Andrews--Tauraso recently introduced an infinite family of $q$-series \[ \mathcal{U}_{t}(a;q):= \sum_{1\le n_1<n_2<\cdots<n_t} \frac{q^{n_1+n_2+\cdots+n_t}}{(1+aq^{n_1}+q^{2n_1})(1+aq^{n_2}+q^{2n_2})\cdots (1+aq^{n_t}+q^{2n_t})} \] and proved that these functions are linear combinations of quasimodular forms. In this paper, we study a broader family of $q$-series that contains the collection $\{\mathcal{U}_t\}_{t \in \mathbb{N}}$. Using the theory of quasi shuffle algebras, we show that this extended family also lies in the algebra of quasimodular forms. Moreover, we determine the precise weights and levels of these functions, thereby making Amdeberhan--Andrews--Tauraso's result sharp. We further investigate the limiting behavior of these functions. In particular, we demonstrate that the sequence of quasimodular forms~$\{\mathcal{U}_t(1;q)\}_{t\in\mathbb{N}}$ gives an approximation for the ordinary partition function. We also establish infinitely many closed formulas for reciprocals of certain infinite products in terms of~$\mathcal{U}_{t}(a;q)$.
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