The Göllnitz-Gordon partition identities are partition identities derived from two analytic series-product identities. \begin{equation}\label{VII.150} \sum_{n=0}^{\infty} \frac{(-q;q^2)_n q^{n^2}}{(q^2;q^2)_{n}}=\frac{1} {(q,q^{4},q^{7};q^{8})_{\infty}}, \end{equation} \begin{equation}\label{VII.160} \sum_{n=0}^{\infty} \frac{(-q;q^2)_n q^{n^2+2n}}{(q^2;q^2)_{n}}=\frac{1} {(q^3,q^{4},q^{5};q^{8})_{\infty}}. \end{equation} The partition interpretation of the first of these identities is the following.

Theorem. The number of partitions of a positive integer $n$ in which the minimal difference between parts is at least 2, and at least 4 between even parts, equals the number of partitions of $n$ into parts congruent to 1, 4, or 7 ( mod 8).

The corresponding interpretation of the second identity is the following.

Theorem. The number of partitions of $n$ 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 $n$ into parts congruent to 3, 4, or 5 (mod 8).

The combinatorial interpretations were found independently by Göllnitz [1] and Gordon [2].
The analytic version of the identities were first published by Lucy Slater in [3].

[1] H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. 225 (1967), 154-190.
[2] B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math J. 32 (1965), 741-748.
[3] L. J. Slater, Further Identities of the Rogers-Ramanujan Type. Proc. London Math. Soc. Ser. 2 54, 147-167, 1952.

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