Multiplication theorem

Remarks

Part 1 is the Gauss multiplication formula, and part 2 follows from it by taking the logarithmic derivative. Part 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 $m$. Part 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 $t/m$. For $m = 2$, part 1 is the Legendre duplication formula, and part 3 gives $B_n(2x) = 2^{n-1}\big(B_n(x) + B_n(x + \tfrac{1}{2})\big)$. At $x = 0$ and $n = 1$ this reads $B_1(0) = B_1(0) + B_1(\tfrac{1}{2})$, consistent with $B_1(\tfrac{1}{2}) = 0$.

Going deeper

For a student

All these functions satisfy a simple difference equation in $x$, such as $\Gamma(x + 1) = x\Gamma(x)$ or $B_n(x + 1) - B_n(x) = n x^{n-1}$. The sum or product over the $m$ shifted points satisfies the same kind of equation in the variable $mx$. A uniqueness argument for that difference equation identifies it with the function at $mx$. For the gamma function this uniqueness is the Bohr–Mollerup theorem. The formulas are used to compute special values, such as $\Gamma(1/2) = \sqrt{\pi}$, and they connect to distribution relations in number theory.