  A Goellnitz-Gordon partition is one in which the parts differ by at least 2,
and where the inequality is strict if a part is even. Let Q_i(n) denote the
number of partitions of n into distinct parts not congruent to i mod 4. By
attaching weights which are powers of 2 and imposing certain parity conditions
on Goellnitz-Gordon partitions, we show that these are equinumerous with Q_i(n)
for i=0,2. These complement results of Goellnitz on Q_i(n) for i=1,3, and of
Alladi who provided a uniform treatment of all four Q_i(n), i=0,1,2,3, in terms
of weighted partitions into parts differing by >= 4. Our approach here provides
a uniform treatment of all four Q_i(n) in terms of certain double series
representations. These double series identities are part of a new infinite
hierarchy of multiple series identities.
