  Liu established an addition formula for the Jacobian theta function by using
the theory of elliptic functions. From this addition formula he obtained the
Ramanujan cubic theta function identity, Winquist's identity, a theta function
identities with five parameters, and many other interesting theta function
identities. In this paper we will give an addition formula for the Jacobian
theta function which is equivalent to Liu's addition formula. Based on this
formula we deduce some known theta function identities as well as new
identities. From these identities we shall establish certain new series
expansions for
