发表机构
CNRS, IMJ-PRG, Sorbonne; Dalhousie University; The University of Melbourne; Barry University; University of Liège（法国国家科学研究中心，数学与交叉研究研究所，索邦大学; 达尔豪斯大学; 墨尔本大学; 巴里大学; 列日大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推广 Gosper 和 Ramanujan 的无穷乘积求值，引入新方法证明含三个自由参数的推广公式，并扩展到高阶多项式情形，同时建立与 Kurokawa–Rovinski 乘积的联系。
AI 中文摘要
Gosper 引入了许多涉及无穷乘积的显著公式。其中一个公式给出了乘积 $$\prod_{n \geq 1} \frac{1}{e}\left(\frac{1}{3n} + 1\right)^{3n + \frac{1}{2}}$$ 的求值。该求值的完整证明似乎尚未出现在文献中。本文中，我们引入一些技术来推广 Gosper 的求值。我们证明了涉及三个自由参数的 Gosper 公式的推广，该推广也可用于求值 Gosper 乘积的任意多分（multisections）。随后，我们研究涉及 $n$ 中更高次多项式的高阶推广，相对于 Gosper 乘积中涉及的多项式 $3n$ 和 $3n+\frac{1}{2}$，从而得到 Ramanujan 引入且近期由 Bradley–Thrush [\textit{\it Ramanujan J.} (2025)] 研究的无穷乘积求值的一个推广。我们还考虑了与 Kurokawa 和 Rovinski\u 的无穷乘积的联系，基于第一和第四作者 [\textit{\it Ramanujan J.} (2025)] 的近期工作。
英文摘要
Gosper introduced many remarkable formulas involving infinite products. One such formula provides an evaluation of the product $$\prod_{n \geq 1} \frac{1}{e}\left(\frac{1}{3n} + 1\right)^{3n + \frac{1}{2}}. $$ It appears that a complete proof of this evaluation has not appeared in the literature. In this paper, we introduce some techniques to extend Gosper's evaluation. We prove a generalization of Gosper's formula involving three free parameters, which can also be applied to evaluate arbitrary multisections of Gosper's product. Then we investigate higher-order extensions involving higher-degree polynomials in $n$, relative to the polynomials $3n$ and $3n+\frac{1}{2}$ involved in Gosper's product, yielding a generalization of an infinite product evaluation due to Ramanujan and recently studied by Bradley--Thrush [\textit{\it Ramanujan J.} (2025)]. We also consider connections with infinite products due to Kurokawa and Rovinski\uı, building on the recent work of the first and fourth authors [\textit{\it Ramanujan J.} (2025)].
Comments37 pages
