ON CERTAIN q-TRIGONOMETRIC IDENTITIES ANALOGOUS TO THAT OF GOSPER’S

Abstract

In “Experiments and discoveries in $q$-trigonometry”, W. Gosper defined the $q$-analogues ${\text{sin}}_q(z)$ and ${\text{cos}}_q(z)$ of $\text{sin }z$ and $\text{cos }z$ respectively. He also conjectured identities for ${\text{sin}}_q(2z),{\text{sin}}_q(3z)$ and ${\text{sin}}_q(5z)$. 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.