$$\eqalign{ & \prod\limits_{k = 1}^{ + \infty } {{{\left( {1 + \frac{1}{{2k}}} \right)}^{{{\left( { - 1} \right)}^k}}}} = \frac{4}{{{\rm B}\left( {\frac{1}{2},\frac{1}{4}} \right)}} \cr & \prod\limits_{k = 1}^{ + \infty } {{{\left( {1 - \frac{1}{{2k}}} \right)}^{{{\left( { - 1} \right)}^k}}}} = \frac{1}{\pi }{\rm B}\left( {\frac{1}{2},\frac{1}{4}} \right) \cr & \prod\limits_{k = 1}^{ + \infty } {\left( {1 + \frac{{{{\left( { - 1} \right)}^{k + 1}}}}{{2k}}} \right)} = \frac{1}{{2\pi }}{\rm B}\left( {\frac{1}{4},\frac{1}{4}} \right) \cr} $$