Some $q$-identities derived by the ordinary derivative operator
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- by Jin Wang and Ruiqi Ruan;
- Proc. Amer. Math. Soc. 152 (2024), 3451-3465
- DOI: https://doi.org/10.1090/proc/16817
- Published electronically: June 21, 2024
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Abstract:
In this paper, we investigate applications of the ordinary derivative operator, instead of the $q$-derivative operator, to the theory of $q$-series. As main results, many new summation and transformation formulas are established which are closely related to some well-known formulas such as the $q$-binomial theorem, Ramanujan’s ${}_1\psi _1$ formula, the quintuple product identity, Gasper’s $q$-Clausen product formula, and Rogers’ ${}_6\phi _5$ 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.References
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Bibliographic Information
- Jin Wang
- Affiliation: School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, People’s Republic of China
- MR Author ID: 1180622
- Email: jinwang@zjnu.edu.cn
- Ruiqi Ruan
- Affiliation: School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, People’s Republic of China
- ORCID: 0009-0003-1804-7384
- Email: ruiqiruan@zjnu.edu.cn
- Received by editor(s): November 29, 2023
- Received by editor(s) in revised form: January 3, 2024, January 23, 2024, and January 27, 2024
- Published electronically: June 21, 2024
- Additional Notes: The first author is the corresponding author
This research was supported by the Natural Science Foundation of Zhejiang Province under Grant No. LY24A010012 and the National Natural Science Foundation of China under Grant No. 12001492. - Communicated by: Mourad Ismail
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3451-3465
- MSC (2020): Primary 33D15; Secondary 05A30
- DOI: https://doi.org/10.1090/proc/16817
- MathSciNet review: 4767275