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 π ) ( m − 1 ) / 2 m 1 / 2 − m z Γ ( m z ) = Γ ( z ) Γ ( z + 1 m ) ⋯ Γ ( z + m − 1 m ) {(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 ( mod b i ) : 1 ⩽ i ⩽ m } \{ {a_i}(\bmod {b_i}):1 \leqslant i \leqslant m\} be an exact covering system with standardized offsets. Then \[ Γ ( z ) = Γ ( z / b 1 ) b 1 1 − z / b 1 ∏ i = 2 m Γ ( ( z + a i ) / b i ) b i − z / b i Γ ( a i / b i ) . \Gamma (z) = \frac {{\Gamma (z/{b_1})}} {{{b_1}^{1 - z/{b_1}}}}\prod \limits _{i = 2}^m {\frac {{\Gamma ((z + {a_i})/{b_i})}} {{b_i^{ - z/{b_i}}\Gamma ({a_i}/{b_i})}}}. \] Conversely, if the above identity holds, then { a i ( mod b i ) : 1 ⩽ i ⩽ m } \{ {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.