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  "body": "\n## Definition\n\nLet $a$ and $q$ be [[complex numbers|09FTAET5Z4]], and let $n \\ge 0$ be an integer.\nThe $q$-Pochhammer symbol is\n$$(a; q)_n = \\prod_{k=0}^{n-1} (1 - a q^k),$$\nwith $(a; q)_0 = 1$.\nIf $\\abs{q} \u003c 1$, the infinite $q$-Pochhammer symbol is the infinite product\n$$(a; q)_\\infty = \\prod_{k=0}^{\\infty} (1 - a q^k).$$\n\n## Remarks\n\nFor $\\abs{q} \u003c 1$ the product defining $(a; q)_\\infty$ converges, since $\\sum_k \\abs{a q^k} = \\abs{a} / (1 - \\abs{q}) \u003c \\infty$; it is zero exactly when some factor is zero.\nExample: $(q; q)_2 = (1 - q)(1 - q^2) = 1 - q - q^2 + q^3$, by the definition with $a = q$ and $n = 2$.\nNon-example: for $\\abs{q} \u003e 1$ and $a \\ne 0$, the factors $1 - a q^k$ do not tend to $1$, so the infinite product does not [[converge|H1Z2TKY3EQ]] and $(a; q)_\\infty$ is not defined.\nThe name comes from the limit $\\frac{(q^x; q)_n}{(1 - q)^n} \\to x(x+1)\\cdots(x+n-1)$ as $q \\to 1$, the rising factorial or Pochhammer symbol.\nThe [[generating function for partitions|0578PA53VA]] reads $\\sum_{n \\ge 0} p(n) q^n = 1/(q; q)_\\infty$ for $\\abs{q} \u003c 1$.\n\n## Explanations\n\n### L1 human\n\nThe symbol $(a; q)_n$ is shorthand for the product of $n$ factors $1 - a$, $1 - aq$, up to $1 - aq^{n-1}$.\n",
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