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March 2022 On Gosper’s $\Pi_q$ and Lambert series identities
Yathirajsharma Mudumbai Varada, Harshitha Kempaiahnahundi Nanjundegowda, Vasuki Kaliyur Ranganna
Author Affiliations +
Hiroshima Math. J. 52(1): 113-137 (March 2022). DOI: 10.32917/h2021044

Abstract

In an interesting article entitled “Experiments and discoveries in $q$-trigonometry”, R. W. Gosper conjectured few beautiful $\Pi_q$ and Lambert series identities. Many people have attempted confirming some of those identities in the Gosper’s list, mainly by using Gosper’s $q$-trigonometric identities. In this paper we either prove or disprove all the $\Pi_q$ and Lambert series identities in the Gosper’s list by mainly using S. Ramanujan’s theta function identities and W. N. Bailey’s summation formula. In the process, we obtain three new Gosper kind of identities.

Funding Statement

The first author is supported by grant No. 09/119(0221)/2019-EMR-1 by the funding agency CSIR, INDIA, under CSIR-JRF.
The second author is supported by grant UGC-Ref. No.: 982/(CSIR-UGC NET DEC.2017) by the funding agency UGC, INDIA, under CSIR-UGC JRF.
The third author is supported by grant No. F. 510/12/DRS-II/2018 (SAP-I) by the University Grants Commission, India.

Acknowledgment

The first author is supported by grant No. 09/119(0221)/2019-EMR-1 by the funding agency CSIR, INDIA, under CSIR-JRF. The second author is supported by grant UGC-Ref. No.: 982/(CSIR-UGC NET DEC.2017) by the funding agency UGC, INDIA, under CSIR-UGC JRF. The third author is supported by grant No. F. 510/12/DRS-II/2018 (SAP-I) by the University Grants Commission, India.

Citation

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Yathirajsharma Mudumbai Varada. Harshitha Kempaiahnahundi Nanjundegowda. Vasuki Kaliyur Ranganna. "On Gosper’s $\Pi_q$ and Lambert series identities." Hiroshima Math. J. 52 (1) 113 - 137, March 2022. https://doi.org/10.32917/h2021044

Information

Received: 2 June 2021; Revised: 11 December 2021; Published: March 2022
First available in Project Euclid: 25 March 2022

Digital Object Identifier: 10.32917/h2021044

Subjects:
Primary: 11F20 , 11M36 , 33C75

Keywords: constant $\Pi_q$ , Eisenstein series , Lambert series , modular equations , Ramanujan theta function

Rights: Copyright © 2022 Hiroshima University, Mathematics Program

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Vol.52 • No. 1 • March 2022
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