Elementary derivations of the Rogers-Fine identity and other q-series identities

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-01-09 DOI:10.1016/j.disc.2024.114387

Heng Huat Chan , Song Heng Chan , Zhi-Guo Liu

引用次数: 0

Abstract

We begin the article with a proof of the Rogers-Fine identity. We then show that the Rogers-Fine identity implies the Rogers-Ramanujan identities as well as a new finite version of the quintuple identity. Motivated by the connections between these identities, we discover an identity which yields proofs of Rogers-Ramanujan-type identities associated with the Rogers-Ramanujan continued fraction, the Ramanujan-Göllnitz-Gordon continued fraction and Ramanujan's cubic continued fraction. We also discover a new generalization of the quintuple product identity which leads to a generalization of an identity due to R.J. Evans and a short proof of q-Chu-Vandermonde identity that does not require the knowledge of the q-binomial theorem.

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文章一开始，我们首先证明了罗杰斯-菲恩同一性。然后，我们证明罗杰斯-菲尼特征蕴含罗杰斯-拉玛努扬特征以及新的有限版本的五元特征。受这些等式之间联系的启发，我们发现了一个等式，它可以证明与罗杰斯-拉玛努扬续分数、拉玛努扬-贡尼茨-戈登续分数和拉玛努扬立方续分数相关的罗杰斯-拉玛努扬型等式。我们还发现了五乘积同一性的新概括，它导致了 R.J. Evans 提出的同一性的概括，以及 q-Chu-Vandermonde 同一性的简短证明，它不需要 q-二项式定理的知识。

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

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.