Exact covering systems and the Gauss-Legendre multiplication formula for the gamma function

Abstract:

The Gauss-Legendre multiplication formula for the gamma function is ${(2\pi )^{(m - 1)/2}}{m^{1/2 - mz}}\Gamma (mz) = \Gamma (z)\Gamma (z + \tfrac {1} {m}) \cdots \Gamma (z + \tfrac {{m - 1}} {m})$. Let $\{ {a_i}(\bmod {b_i}):1 \leqslant i \leqslant m\}$ be an exact covering system with standardized offsets. Then \[ \Gamma (z) = \frac {{\Gamma (z/{b_1})}} {{{b_1}^{1 - z/{b_1}}}}\prod \limits _{i = 2}^m {\frac {{\Gamma ((z + {a_i})/{b_i})}} {{b_i^{ - z/{b_i}}\Gamma ({a_i}/{b_i})}}}.\] Conversely, if the above identity holds, then $\{ {a_i}(\bmod {b_i}):1 \leqslant i \leqslant m\}$ is an exact covering system with standardized offsets. The Gauss-Legendre multiplication formula is a special case of this identity.