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


Author: John Beebee
Journal: Proc. Amer. Math. Soc. 120 (1994), 1061-1065
MSC: Primary 33B15
DOI: https://doi.org/10.1090/S0002-9939-1994-1180463-4
MathSciNet review: 1180463
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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

$\displaystyle \Gamma (z) = \frac{{\Gamma (z/{b_1})}} {{{b_1}^{1 - z/{b_1}}}}\pr... ...{\frac{{\Gamma ((z + {a_i})/{b_i})}} {{b_i^{ - z/{b_i}}\Gamma ({a_i}/{b_i})}}}.$

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.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1180463-4
Article copyright: © Copyright 1994 American Mathematical Society

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