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  "body": "\n## Summary\n\nThe quantum dilogarithm is a [[q-analog|GFSDMJHHVA]] of the [[dilogarithm|G0X0NWSDB7]].\nIn its basic form it is the infinite product $(x; q)_\\infty = \\prod_{n \\ge 0} (1 - x q^n)$, whose logarithm is a deformed dilogarithm series.\nIt satisfies a pentagon identity for $q$-commuting variables, a quantum version of the five-term relation.\nIt appears in quantum groups, quantum Teichmüller theory, cluster algebras and knot invariants.\n\n## Explanations\n\n### L1 human\n\nThe ordinary dilogarithm is the series $\\sum_{n \\ge 1} x^n / n^2$.\nReplace one factor $n$ in the denominator by the \"$q$-number\" $1 - q^n$, and you get the series $\\sum_{n \\ge 1} x^n / \\big(n (1 - q^n)\\big)$.\nIts exponential is a simple product, $\\prod_{n \\ge 0} (1 - x q^n)^{-1}$.\nAs $q$ tends to $1$, the deformed series times $1 - q$ tends back to the dilogarithm, so the classical function is a limit of the quantum one.\nFor example, with $q = 0$ the product $(x; q)_\\infty$ is just $1 - x$, since every factor after the first equals $1$.\n",
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