On series identities of Gosper and integrals of Ramanujan theta function 𝜓(𝑞)

Abstract

We prove some Lambert series identities which were stated by Gosper without proof or reference. As an application we shall evaluate integrals involving Ramanujan theta function ψ ( q ) \psi (q) . Furthermore, motivated by Ramanujan identities for q ψ 4 ( q 2 ) q\psi ^4(q^2) and ψ 3 ( q ) ψ ( q 3 ) \frac {\psi ^3(q)}{\psi (q^3)} , we shall evaluate the squares of q ψ 4 ( q 2 ) q\psi ^4(q^2) and ψ 3 ( q ) ψ ( q 3 ) \frac {\psi ^3(q)}{\psi (q^3)} in terms of Lambert series.