The interplay between the theory of basic hypergeometric functions and the theory of integer partitions has a long history, going back at least as far as Leonhard Euler [1].

The most obvious connection between basic hypergeometric series and partitions is that the generating function the regular partition function, and various restricted partition functions, are expressable in various ways as $q$-series or $q$-products.

Identities relating various basic hypergeometric series and products of the form \[ A(q)=B(q) \] may be proved combinarorially, by interpreting $A(q)$ and $B(q)$ as generating functions for sequences $\{a(n)\}$ and $\{b(n)\}$, \[ A(q)=\sum_{n=0}^{\infty}a(n)q^n, \qquad B(q)=\sum_{n=0}^{\infty}b(n)q^n, \] where $a(n)$ and $b(n)$ count certain kinds of partitions of the integer $n$, and then showing that $a(n)=b(n)$ for all $n$ by exhibiting a bijection between the two types of partitions. Identities for basic hypergeometric series and/or products with several variables may be proved similarly.

For example, it was shown on the Combinatorial Proofs of Partition Identities page that if $a(m,n)$ denotes the number of self-conjugate partitions of the integer $n$ with Durfee square of side $m$ and $b(m,n)$ denotes the number of partitions of $n$ into $m$ distinct odd parts, then $a(m,n)=b(m,n)$ for all integers $m$ and $n$. Since \[ \sum_{m,n=0}^{\infty}a(m,n)z^mq^n = \sum_{k=0}^{\infty}\frac{z^kq^{k^2}}{\prod_{j=1}^k (1-q^{2j})}, \qquad \sum_{m,n=0}^{\infty}b(m,n)z^mq^n=\prod_{j=1}^{\infty}(1+z q^{2j-1}), \] this shows that \[ \sum_{k=0}^{\infty}\frac{z^kq^{k^2}}{\prod_{j=1}^k(1-q^{2j})}=\prod_{j=1}^{\infty}(1+z q^{2j-1}). \]

In the other direction of course, an analytic identity between two basic hypergeometric functions often implies a partition identity, as described on the Analytic Proofs of Partition Identities page.

[E1] L. Euler, Introductio in Analysin Infinitorum, MarcunMichaelem Bousquet, Lausanne (1748). (link)

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