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

## TL;DR

This paper proves new Lambert series identities originally stated by Gosper, and applies these results to evaluate integrals involving Ramanujan's theta function, as well as related squared expressions inspired by Ramanujan's identities.

## Contribution

It introduces new Lambert series identities related to Ramanujan theta functions and evaluates complex integrals and squared expressions based on these identities.

## Key findings

- Proved Lambert series identities originally stated by Gosper.
- Evaluated integrals involving Ramanujan theta function $\psi(q)$.
- Expressed squares of certain Ramanujan theta function combinations in terms of Lambert series.

## 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 $\fr{\psi^3(q)}{\psi(q^3)}$, we shall evaluate the squares of $q\psi^4(q^2)$ and $\fr{\psi^3(q)}{\psi(q^3)}$ in terms of Lambert series.