A THREE TERM THETA FUNCTION IDENTITY AND ITS APPLICATIONS

Abstract. In this paper we establish a three term theta function iden-tity using the theory of complex variable theory of elliptic functions. This simple identity in form turns out to be quite useful and it is a com-mon origin of many important theta function identities. From which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other inter-esting theta function identities are derived. We also give a new proof of the addition theorem for the Weierstrass elliptic function ℘. An identity involving the products of four theta functions is given and from which one theta function identity by S.McCullough and L.-C. Shen is derived. The quintuple product identity is used to derive some Eisenstein series identities found in Ramanujan’s lost notebook and our approach is dif-ferent from that of B.C. Berndt and Ae Ja Yee. The proofs are self contained and elementary. 1