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  "body": "\n## Statement\n\nLet $\\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\\C$, and let $m$ be a positive integer.\nFor every $z \\in \\C$ with $mz \\notin \\set{0, -1, -2, \\ldots}$,\n$$\\prod_{k=0}^{m-1} \\Gamma\\!\\left(z + \\frac{k}{m}\\right) = (2\\pi)^{(m-1)/2} \\, m^{1/2 - mz} \\, \\Gamma(mz),$$\nwhere $m^{1/2 - mz} = e^{(1/2 - mz)\\ln m}$.\n\n## Remarks\n\nFor $x \u003e 0$, the function $f(x) = \\frac{m^{x - 1/2}}{(2\\pi)^{(m-1)/2}} \\prod_{k=0}^{m-1} \\Gamma\\!\\left(\\frac{x + k}{m}\\right)$ satisfies $f(x+1) = x f(x)$ and $\\ln f$ is convex.\nSo $f = c\\,\\Gamma$ for a constant $c$, by the argument of the [[Bohr–Mollerup theorem|QFF87AXWN2]]; the constant is found from $f(1) = 1$, which uses $\\prod_{k=1}^{m-1}\\Gamma(k/m) = (2\\pi)^{(m-1)/2}/\\sqrt{m}$, a consequence of the [[reflection formula|WF7KQAQTS2]].\nAlternatively, the constant follows from [[Stirling's formula|53750WSJ6K]] as $z \\to \\infty$.\nFor $m = 1$ the formula is trivial, and for $m = 2$ it is the [[Legendre duplication formula|CND4E1PY9V]], since $(2\\pi)^{1/2} 2^{1/2 - 2z} = 2^{1-2z}\\sqrt{\\pi}$.\n\n## Explanations\n\n### L1 human\n\nThe multiplication formula splits $\\Gamma$ at $m$ times a number into a product of $m$ values of $\\Gamma$ at evenly spaced points.\nIt generalizes the duplication formula from $2$ to any whole number $m$.\n\n### L2 human\n\nTaking logarithms and differentiating gives the matching multiplication formula for the [[digamma function|HWRRXB0YXS]].\nSimilar multiplication theorems hold for other special functions, such as the Hurwitz zeta function and the Bernoulli polynomials.\nSetting $z = 1/m$ gives the product $\\prod_{k=1}^{m-1}\\Gamma(k/m) = (2\\pi)^{(m-1)/2}/\\sqrt{m}$.\n",
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