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

Abstract

We prove some Lambert series 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's identities for $q{\psi }^4(q^2)$ and $\fr{ψ^3(q)}{ψ(q^3)}$, we shall evaluate the squares of $q{\psi }^4(q^2)$ and $\fr{ψ^3(q)}{ψ(q^3)}$ in terms of Lambert series.

arXiv comment13 pages