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
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
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