  By considering a limiting form of the q-Dixon_4\phi_3 summation, we prove a
weighted partition theorem involving odd parts differing by >= 4. A two
parameter refinement of this theorem is then deduced from a quartic
reformulation of Goellnitz's (Big) theorem due to Alladi, and this leads to a
two parameter extension of Jacobi's triple product identity for theta
functions. Finally, refinements of certain modular identities of Alladi
connected to the Goellnitz-Gordon series are shown to follow from a limiting
form of the q-Dixon_4\phi_3 summation.
