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  "body": "\n## Summary\n\nA q-analog of a formula or a theorem is a version with an extra parameter $q$, see the [[definition|CK80HQNHEK]].\nIt gives back the original in the limit $q \\to 1$.\nThe basic example is the q-integer $[n]_q = 1 + q + \\cdots + q^{n-1}$, which tends to $n$.\nFrom it one builds q-factorials, [[Gaussian binomial coefficients|YS09Z1YNEK]] and q-series.\nGood q-analogs often have a meaning of their own, such as counts over [[finite fields|0ZXZ82FTX2]] or generating functions of statistics on [[permutations|05RWFKVTJ7]].\n\n## Explanations\n\n### L1 human\n\nTake a familiar number, such as $3$, and replace it by the polynomial $1 + q + q^2$.\nSetting $q = 1$ gives back $3$, so nothing is lost.\nBut the polynomial carries more information: for instance, it can count objects by size and by an extra feature at the same time.\nA q-analog of a theorem is a statement about such polynomials that becomes the original theorem when $q = 1$.\n",
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