Septic theta function identities in Ramanujan's lost notebook (Q2717635)

Let \(\varphi(q)\) be the theta series \(\sum _{k=-\infty} ^{\infty}q^{k^2}\). In his so-called ``Lost Notebook'', Ramanujan records an identity NEWLINE\[NEWLINE\frac {\varphi(q^{1/7})} {\varphi(q^{7})} =1+u+v+w,NEWLINE\]NEWLINE where \(u,v,w\) are implicitly given by certain algebraic equations. Explicit expressions for \(u,v,w\) can be extracted by comparison with an entry in Ramanujan's Second Notebook. The identity together with the algebraic equations are proven in this paper, by employing modular equations of degree seven, the Jacobi triple product identity, several Lambert series identities and a recent product formula for theta functions due to the author.