On the equivalence of Ramanujan's partition identities and a connection with the Rogers-Ramanujan continued fraction
The author uses the Dedekind \(\eta\) transformation formula to prove the equivalence of two identities of Ramanujan: \[ \sum^\infty_{n=1} \left({n \over 5} \right) {q^n \over (1-q^n)^2} = q{(q^5;q^5)^5_\infty \over (q;q)_\infty} \text{ and } 1-5\sum^\infty_{n=1} \left( {n\over 5} \right) {nq^n \over 1-q^n} = {(q;q)^5_\infty \over (q^5; q^5)_\infty}, \] where \((n/5)\) is the Legendre symbol. He also proves the equivalence of two identities of Ramanujan that are related to the modulus 7 and shows that the second of the identities given above implies the differential equation involving the Rogers-Ramanujan continued fraction.
scientific article; zbMATH DE number 902490