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  "body": "\n## Statement\n\nFor an integer $n \\ge 0$, let $(q; q)_n = (1 - q)(1 - q^2) \\cdots (1 - q^n)$, the [[q-Pochhammer symbol|Y1X8ZZQ7VM]], with $(q; q)_0 = 1$.\nIn the ring of [[formal power series|033WGN00BF]] $\\Z[\\![q]\\!]$, the following two identities hold.\n\n1. $$\\sum_{n=0}^{\\infty} \\frac{q^{n^2}}{(q; q)_n} = \\prod_{n=0}^{\\infty} \\frac{1}{(1 - q^{5n+1})(1 - q^{5n+4})}.$$\n2. $$\\sum_{n=0}^{\\infty} \\frac{q^{n^2 + n}}{(q; q)_n} = \\prod_{n=0}^{\\infty} \\frac{1}{(1 - q^{5n+2})(1 - q^{5n+3})}.$$\n\n## Remarks\n\nIn terms of [[integer partitions|0JV30XN73Z]], the identities say the following for each $m \\ge 0$.\nThe partitions of $m$ whose parts differ pairwise by at least $2$ are as many as the partitions of $m$ into parts congruent to $1$ or $4$ modulo $5$.\nConsider the partitions of $m$ whose parts differ by at least $2$ and are all at least $2$.\nThey are as many as the partitions of $m$ into parts congruent to $2$ or $3$ modulo $5$.\nThe term $q^{n^2} / (q; q)_n$ counts the partitions of the first kind with exactly $n$ parts, since $n^2 = 1 + 3 + \\cdots + (2n - 1)$ is the smallest such sum.\n\nExample: for $m = 6$, the partitions with parts differing by at least $2$ are $6$, $5 + 1$ and $4 + 2$.\nThe partitions into parts from $\\set{1, 4, 6, 9, \\ldots}$ are $6$, $4 + 1 + 1$ and $1 + 1 + 1 + 1 + 1 + 1$.\nBoth counts are $3$.\n\nA standard proof transforms the sum side and then applies the Jacobi triple product to reach the product side.\nThe identities were found by Leonard James Rogers and later found again by Srinivasa Ramanujan.\n\n## Explanations\n\n### L1 human\n\nCount the ways to write a number as a sum of parts that are far apart, each two parts differing by at least $2$.\nCount also the ways to write it as a sum of parts that end in $1, 4, 6$ or $9$.\nThe theorem says that the two counts always agree.\nNothing in the two rules suggests the connection, which is why the identities are famous.\n\n### L2 human\n\nThe identities are the prototype of partition identities of Rogers–Ramanujan type, where a difference condition on the parts matches a congruence condition.\nTheir sum sides are basic hypergeometric series and their product sides are, up to a power of $q$, quotients of theta functions.\nThey appear in the theory of modular forms, in the hard hexagon model of statistical mechanics, and in the representation theory of affine Lie algebras.\nCompare with [[Euler's partition theorem|0XFZ1BMF5Q]], where parts are distinct on one side and odd on the other.\n",
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