Some 𝑞-identities derived by the ordinary derivative operator

Pub Date : 2024-02-29 DOI:10.1090/proc/16817

Jin Wang, Ruiqi Ruan

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引用次数: 0

Abstract

In this paper, we investigate applications of the ordinary derivative operator, instead of the q q -derivative operator, to the theory of q q -series. As main results, many new summation and transformation formulas are established which are closely related to some well-known formulas such as the q q -binomial theorem, Ramanujan’s 1 ψ 1 {}_1\psi _1 formula, the quintuple product identity, Gasper’s q q -Clausen product formula, and Rogers’ 6 ϕ 5 {}_6\phi _5 formula, etc. Among these results is a finite form of the Rogers-Ramanujan identity and a short way to Eisenstein’s theorem on Lambert series.

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由普通导数算子推导出的一些Δ常数

本文研究了普通导数算子而非 q q -导数算子在 q q -数列理论中的应用。作为主要成果，我们建立了许多新的求和与变换公式，它们与一些著名公式密切相关，如 q q -二项式定理、Ramanujan 的 1 ψ 1 {}_1\psi _1 公式、五次乘积同一性、Gasper 的 q q -Clausen 乘积公式和 Rogers 的 6 ϕ 5 {}_6\phi _5 公式等。在这些结果中，有罗杰斯-拉马努扬特性的有限形式，也有爱森斯坦兰伯特级数定理的捷径。

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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