Remarks on MacMahon's q-series

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A Pub Date : 2024-06-03 DOI:10.1016/j.jcta.2024.105921

Ken Ono, Ajit Singh

引用次数: 0

Abstract

In his important 1920 paper on partitions, MacMahon defined the partition generating functions $A_{k} (q) = \sum_{n = 1}^{\infty} m (k; n) q^{n} : = \sum_{0 < s_{1} < s_{2} < \dots < s_{k}} \frac{q^{s_{1} + s_{2} + \dots + s_{k}}}{{(1 - q^{s_{1}})}^{2} {(1 - q^{s_{2}})}^{2} \dots {(1 - q^{s_{k}})}^{2}},$ $C_{k} (q) = \sum_{n = 1}^{\infty} m_{odd} (k; n) q^{n} : = \sum_{0 < s_{1} < s_{2} < \dots < s_{k}} \frac{q^{2 s_{1} + 2 s_{2} + \dots + 2 s_{k} - k}}{{(1 - q^{2 s_{1} - 1})}^{2} {(1 - q^{2 s_{2} - 1})}^{2} \dots {(1 - q^{2 s_{k} - 1})}^{2}} .$ These series give infinitely many formulas for two prominent generating functions. For each non-negative k, we prove that $A_{k} (q), A_{k + 1} (q), A_{k + 2} (q), \dots$ (resp. $C_{k} (q), C_{k + 1} (q), C_{k + 2} (q), \dots$ ) give the generating function for the 3-colored partition function $p_{3} (n)$ (resp. the overpartition function $\overline{p} (n)$ ).

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关于麦克马洪 Q 系列的评论

在 1920 年关于分区的重要论文中，麦克马洪定义了分区生成函数Ak(q)=∑n=1∞m(k;n)qn：=∑0<s1<s2<⋯<skqs1+s2+⋯+sk(1−qs1)2(1−qs2)2⋯(1−qsk)2,Ck(q)=∑n=1∞modd(k;n)qn:=∑0<s1<s2<⋯<skq2s1+2s2+⋯+2sk−k(1−q2s1−1)2(1−q2s2−1)2⋯(1−q2sk−1)2.这些数列给出了两个著名生成函数的无限多公式。对于每个非负 k，我们证明 Ak(q),Ak+1(q),Ak+2(q),...（即 Ck(q),Ck+1(q),Ck+2(q),...）给出了三色分割函数 p3(n)（即超分割函数 p‾(n)）的生成函数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Combinatorial Theory Series A 数学-数学

CiteScore

2.90

自引率

9.10%

发文量

审稿时长

12 months

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.