A limiting form of the q-Dixon₄ϕ₃ summation and related partition identities

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By considering a limiting form of the q-Dixon₄ϕ₃ 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-Dixon₄ϕ₃ summation.