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On series identities of Gosper and integrals of Ramanujan theta function $ \psi(q)$


Author: Mohamed El Bachraoui
Journal: Proc. Amer. Math. Soc. 147 (2019), 4451-4464
MSC (2010): Primary 33E05, 11F11, 11F12
DOI: https://doi.org/10.1090/proc/14690
Published electronically: June 14, 2019
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Abstract: We prove some Lambert series identities which were stated by Gosper without proof or reference. As an application we shall evaluate integrals involving Ramanujan theta function $ \psi (q)$. Furthermore, motivated by Ramanujan identities for $ q\psi ^4(q^2)$ and $ \frac {\psi ^3(q)}{\psi (q^3)}$, we shall evaluate the squares of $ q\psi ^4(q^2)$ and $ \frac {\psi ^3(q)}{\psi (q^3)}$ in terms of Lambert series.


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Additional Information

Mohamed El Bachraoui
Affiliation: Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al-Ain, United Arab Emirates
Email: melbachraoui@uaeu.ac.ae

DOI: https://doi.org/10.1090/proc/14690
Keywords: Ramanujan theta functions, Lambert series, integrals, $q$-trigonometric functions.
Received by editor(s): January 31, 2019
Published electronically: June 14, 2019
Communicated by: Mourad Ismail
Article copyright: © Copyright 2019 American Mathematical Society