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  "body": "\n## Statement\n\nLet $m \\ge 1$ be an integer.\n\n1. For the [[gamma function|0H5SYBSED9]] and every $z \\in \\C$ with $mz \\notin \\set{0, -1, -2, \\dots}$,\n$$\\prod_{k=0}^{m-1} \\Gamma\\!\\left(z + \\frac{k}{m}\\right) = (2\\pi)^{(m-1)/2}\\, m^{1/2 - mz}\\, \\Gamma(mz).$$\n2. For the [[digamma function|HWRRXB0YXS]] $\\psi = \\Gamma'/\\Gamma$ and the same $z$,\n$$\\psi(mz) = \\ln m + \\frac{1}{m} \\sum_{k=0}^{m-1} \\psi\\!\\left(z + \\frac{k}{m}\\right).$$\n3. For the [[Bernoulli polynomials|BXQKBJJGFG]] $B_n$, $n \\ge 0$, and every $x \\in \\C$,\n$$B_n(mx) = m^{n-1} \\sum_{k=0}^{m-1} B_n\\!\\left(x + \\frac{k}{m}\\right).$$\n4. For the [[Hurwitz zeta function|G4SW7HJG88]] $\\zeta(s, a) = \\sum_{j \\ge 0} (j + a)^{-s}$, every real $x \u003e 0$ and every $s$ with $\\operatorname{Re} s \u003e 1$,\n$$\\sum_{k=0}^{m-1} \\zeta\\!\\left(s, x + \\frac{k}{m}\\right) = m^{s} \\zeta(s, mx).$$\n\n## Remarks\n\nPart 1 is the [[Gauss multiplication formula|Y4759V8CY6]], and part 2 follows from it by taking the logarithmic derivative.\nPart 4 comes from splitting the sum defining $\\zeta(s, mx) = m^{-s} \\sum_{j \\ge 0} (j/m + x)^{-s}$ according to the residue $k$ of $j$ modulo $m$.\nPart 3 follows from the generating function $t e^{xt}/(e^t - 1) = \\sum_n B_n(x) t^n/n!$: summing over $k$ gives a geometric sum that collapses to the generating function at $mx$ with $t$ replaced by $t/m$.\nFor $m = 2$, part 1 is the Legendre duplication formula, and part 3 gives $B_n(2x) = 2^{n-1}\\big(B_n(x) + B_n(x + \\tfrac{1}{2})\\big)$.\nAt $x = 0$ and $n = 1$ this reads $B_1(0) = B_1(0) + B_1(\\tfrac{1}{2})$, consistent with $B_1(\\tfrac{1}{2}) = 0$.\n\n## Explanations\n\n### L1 human\n\nSeveral classical functions have a rule for what happens when the argument is multiplied by a whole number $m$.\nThe value at $mx$ can be recovered from the values at the $m$ evenly spaced points $x, x + 1/m, \\dots, x + (m-1)/m$.\nThese rules hold for the gamma function, for the Bernoulli polynomials and for the Hurwitz zeta function, among others.\n\n### L2 human\n\nAll these functions satisfy a simple difference equation in $x$, such as $\\Gamma(x + 1) = x\\Gamma(x)$ or $B_n(x + 1) - B_n(x) = n x^{n-1}$.\nThe sum or product over the $m$ shifted points satisfies the same kind of equation in the variable $mx$.\nA uniqueness argument for that difference equation identifies it with the function at $mx$.\nFor the gamma function this uniqueness is the Bohr–Mollerup theorem.\nThe formulas are used to compute special values, such as $\\Gamma(1/2) = \\sqrt{\\pi}$, and they connect to distribution relations in number theory.\n",
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