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Question Number 127512 by mohammad17 last updated on 30/Dec/20

Commented by sts last updated on 30/Dec/20

Answered by Dwaipayan Shikari last updated on 30/Dec/20

$$\beta (n,n)=\frac{{\mathrm{\Gamma }}^2\left(n\right)}{\mathrm{\Gamma }\left(2n\right)}=\frac{{\mathrm{\Gamma }}^2\left(n\right)}{\mathrm{\Gamma }\left(n\right)\mathrm{\Gamma }(n+\frac{1}{2})}{2}^{1-2n}\sqrt{\pi }=\frac{\mathrm{\Gamma }\left(n\right)\mathrm{\Gamma }\left(\frac{1}{2}\right)}{\mathrm{\Gamma }(n+\frac{1}{2})}{2}^{1-2n}$$$$=\beta (n,\frac{1}{2})2^{1-2n}$$$$[As\mathrm{\Gamma }\left(n\right)\mathrm{\Gamma }(n+\frac{1}{2})=2^{1-2n}\sqrt{\pi }\mathrm{\Gamma }\left(2n\right)]$$

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