April 2023 ON CERTAIN q-TRIGONOMETRIC IDENTITIES ANALOGOUS TO THAT OF GOSPER’S
M. V. Yathirajsharma
Rocky Mountain J. Math. 53(2): 629-646 (April 2023). DOI: 10.1216/rmj.2023.53.629

Abstract

In “Experiments and discoveries in q-trigonometry”, W. Gosper defined the q-analogues sinq(z) and cosq(z) of sin z and cos z respectively. He also conjectured identities for sinq(2z),sinq(3z) and sinq(5z). Here, we give a brief account of Gosper’s q-trigonometric functions and build some generalized Gosper’s kind of q-trigonometric identities. As a consequence of which, we build four generalized finite q-trigonometric sums, three of which seem to be new. We also give simple proofs to one of Gosper’s Lambert series identities and two of Ramanujan’s Eisenstein series identities. We make use of the technique of this article to show how some new unusual Eisenstein series identities can be deduced.

Citation

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M. V. Yathirajsharma. "ON CERTAIN q-TRIGONOMETRIC IDENTITIES ANALOGOUS TO THAT OF GOSPER’S." Rocky Mountain J. Math. 53 (2) 629 - 646, April 2023. https://doi.org/10.1216/rmj.2023.53.629

Information

Received: 23 November 2021; Revised: 13 May 2022; Accepted: 22 May 2022; Published: April 2023
First available in Project Euclid: 20 June 2023

MathSciNet: MR4604778
zbMATH: 07725159
Digital Object Identifier: 10.1216/rmj.2023.53.629

Subjects:
Primary: 11F20 , 11M36 , 14K25

Keywords: Eisenstein series , Jacobi theta function , q-trigonometric identity , Ramanujan’s theta function

Rights: Copyright © 2023 Rocky Mountain Mathematics Consortium

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Vol.53 • No. 2 • April 2023
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