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  "body": "\n## Definition\n\nLet $q$ be an indeterminate, or a complex number close to $1$.\nA *q-analog* of an expression $E$ is an expression $E_q$, depending on $q$, such that $\\lim_{q \\to 1} E_q = E$.\nA q-analog of an identity or a theorem is an identity or a theorem about q-analogs that reduces to the original one as $q \\to 1$.\nFor an integer $n \\ge 0$, the *q-integer* is\n$$[n]_q = \\frac{1 - q^n}{1 - q} = 1 + q + \\cdots + q^{n-1},$$\nand the *q-factorial* is $[n]_q! = [1]_q [2]_q \\cdots [n]_q$, with $[0]_q! = 1$.\n\n## Remarks\n\nExample: $[n]_q$ is a q-analog of $n$, since $1 + q + \\cdots + q^{n-1} \\to n$ as $q \\to 1$.\nLikewise $[n]_q!$ is a q-analog of $n!$.\nIt is the generating function of the number of inversions over the [[permutations|0BQMWCNS8S]] of $\\set{1, \\ldots, n}$: $[n]_q! = \\sum_{\\pi} q^{\\operatorname{inv}(\\pi)}$.\nFor $n = 3$, the inversion numbers are $0, 1, 1, 2, 2, 3$, and $[3]_q! = (1)(1 + q)(1 + q + q^2) = 1 + 2q + 2q^2 + q^3$.\n\nNon-example: $q^n$ is not a q-analog of $n$, since $q^n \\to 1$ as $q \\to 1$, and $1 \\ne n$ for $n \\ne 1$.\n\nThe definition by a limit alone does not fix a unique q-analog.\nIn practice a q-analog is chosen because it keeps the structure of the original, such as a recurrence or a combinatorial meaning.\nMain examples are the [[Gaussian binomial coefficients|YS09Z1YNEK]], the [[q-Pochhammer symbol|Y1X8ZZQ7VM]] and the q-binomial theorem.\n",
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