$$\mathop {\lim }\limits_{n \to \infty } \left( {n \cdot \prod\limits_{k = 1}^n {\frac{{1 + {{\left( {2k} \right)}^2}}}{{1 + {{\left( {2k + 1} \right)}^2}}}} } \right) = \tanh \frac{\pi }{2}$$

$$\prod\limits_{k = 1}^{ + \infty } {\frac{{2k\left( {4k + 3} \right)}}{{\left( {2k + 1} \right)\left( {4k + 1} \right)}}} = \frac{1}{6}{\rm B}\left( {\frac{1}{2},\frac{1}{4}} \right)$$

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

$$\prod\limits_{k = 1}^{ + \infty } {\frac{{2k\left( {2k + 2} \right)}}{{{{\left( {2k + 1} \right)}^2}}}} = \frac{\pi }{4}$$

$$\prod\limits_{k = 1}^{ + \infty } {\frac{{{{\left( {3k + 2} \right)}^3}}}{{27k{{\left( {k + 1} \right)}^2}}}} = {\left( {\frac{3}{{2\Gamma \left( {\frac{2}{3}} \right)}}} \right)^3}$$