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  "body": "\n## Summary\n\nA $q$-difference equation relates the values of an unknown function at the points $x, qx, q^2 x, \\dots$, where $q \\ne 0, 1$ is a fixed number.\nIt is a difference equation in which the shift $x \\mapsto x + 1$ is replaced by the dilation $x \\mapsto qx$.\nEquivalently, it can be written with the [[q-derivative|PMJT3FZE0Y]] in place of the ordinary derivative.\nSuch equations govern $q$-analogs of special functions, such as basic hypergeometric series, and they tend formally to differential equations as $q \\to 1$.\n\n## Explanations\n\n### L1 human\n\nAn ordinary difference equation looks at a function at $x$ and at $x + 1$.\nA $q$-difference equation looks at $x$ and at $qx$, a point a fixed factor away.\nThe simplest one is $y(qx) = y(x)$.\nFor $q = 2$, any function of $\\log_2 x$ that has period $1$ solves it, for example $\\cos(2\\pi \\log_2 x)$ for $x \u003e 0$.\n",
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