The first Göllnitz-Gordon identity states that the number of partitions of 
in which the minimal difference between parts is at least 2, and at least 4 between
even parts, equals the number of partitions of 
into parts congruent to 1, 4, or 7 (mod 8). For example, taking
,
the resulting two sets of partitions are 
and 
.
The second Göllnitz-Gordon identity states that the number of partitions of 
in which the minimal difference between parts is at least 2, the minimal difference
between even parts is at least 4, and all parts are greater than 2, equals the number
of partitions of 
into parts congruent to 3, 4, or 5 (mod 8). For example, taking
,
the resulting two sets of partitions are 
and 
.
The Göllnitz-Gordon identities are due to H. Göllnitz and were included in his 1961 unpublished honors baccalaureate thesis. However, essentially no one knew about the results until Gordon (1965) independently rediscovered them.
The analytic counterparts of the Göllnitz-Gordon partition identities are the q-series identities
 |
(OEIS A036016 and A036015), where 
denotes a q-series and the coefficients give
the number of partitions satisfying the corresponding Göllnitz-Gordon identity.
These analytic identities were published by Slater (1952) and predate the partition theorem by a decade. Equation (◇) is number 36 and equation (◇) is number 34 in Slater's list. However, it has recently been discovered by A. Sills that two analytic identities equivalent to the analytic Göllnitz-Gordon identities were recorded by Ramanujan in his lost notebook, and thus that Ramanujan knew these identities more than 30 years before Slater rediscovered them (Andrews and Berndt 2008, p. 37)!
