$ \beta{(m,m)} = \frac{\Gamma{m}\Gamma{m}}{\Gamma{2m}} \\ $

$ \text{ By Duplication Formula,} \\ $

$ \Gamma{m} \Gamma{(m+ \frac{1}{2})} = \frac{\sqrt \pi}{2^{2m-1}} \sqrt{2m} \\ $

$ \frac{\Gamma{m}}{\Gamma{2m}} = \frac{\sqrt \pi}{2^{2m-1}} . \frac{1}{\Gamma{(m+ \frac{1}{2}})} \\ $

$ \beta{(m,m)} = \frac{\sqrt \pi} {2^{2m-1}} \frac{\Gamma m}{\Gamma{(m+ \frac{1}{2}})} \\ $

$ = \frac{1} {2^{2m-1}} \frac{\Gamma m \Gamma \frac{1}{2}}{\Gamma{(m+ \frac{1}{2}})} \\ $

$ = \frac{1} {2^{2m-1}} \beta(m,\frac{1}{2}) \\ $

$ \beta{(m,m)} = 2^{1-2m} \beta(m,\frac{1}{2}) \\ $