An extension of the quintuple product identity and its applications

Using the theory of elliptic theta functions, we establish a theta function identity that may be regarded as an extension of the quintuple identity, with many other results, both classical and new, included as special cases. It allows us to give a new derivation of the Ramanujan-Watson modular equation of the seventh order. We give new proofs of some Eisenstein series identities of Ramanujan related to modular equations of degree 7.