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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|JFP2DGMNXT]] 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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