麦克马洪除数和的扩展

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Tewodros Amdeberhan, George E. Andrews, Roberto Tauraso
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引用次数: 0

摘要

1920 年,麦克马洪(P. A. MacMahon)将除数和(经典)概念与整数分区理论联系起来,对其进行了概括。在本文中,我们扩展了麦克马洪的思想。在此过程中,我们揭示了大量可分性定理和意想不到的组合特性。我们最初的方法与 MacMahon 截然不同,涉及 MacMahon 型生成函数的有理函数近似。其中一个例子涉及多个 q 次谐波和 $$\begin{aligned}\sum _{k=1}^n\frac{(-1)^{k-1}\genfrac[]{0.0pt}{}{n}{k}_{q}(1+q^k)q^{\left( {\begin{array}{c}k\\ 2\end{array}}\right) +tk}}{[k]_q^{2t}\genfrac[]{0.0pt}{}{n+k}{k}_{q}} =sum _{1\le k_1\le \le k_{2t}}\le n}\frac{q^{n+k_1+k_3\cdots +k_{2t-1}}+q^{k_2+k_4+\cdots +k_{2t}}}{[n+k_1]_q[k_2]_q\cdots [k_{2t}]_q}。\end{aligned}$$
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Extensions of MacMahon’s sums of divisors

In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so, we reveal a wealth of divisibility theorems and unexpected combinatorial identities. Our initial approach is quite different from MacMahon and involves rational function approximation to MacMahon-type generating functions. One such example involves multiple q-harmonic sums

$$\begin{aligned} \sum _{k=1}^n\frac{(-1)^{k-1}\genfrac[]{0.0pt}{}{n}{k}_{q}(1+q^k)q^{\left( {\begin{array}{c}k\\ 2\end{array}}\right) +tk}}{[k]_q^{2t}\genfrac[]{0.0pt}{}{n+k}{k}_{q}} =\sum _{1\le k_1\le \cdots \le k_{2t}\le n}\frac{q^{n+k_1+k_3\cdots +k_{2t-1}}+q^{k_2+k_4+\cdots +k_{2t}}}{[n+k_1]_q[k_2]_q\cdots [k_{2t}]_q}. \end{aligned}$$
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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