Evaluating the ratio of beta functions

Question

I came across a question which asks for the value of $\alpha$ in the result to the ratio of two $\mathrm{B}$ functions:

$$\frac{\mathrm{B}(m, \frac{1}{2})}{\mathrm{B}(m, m)}=2^{\alpha}$$

I know the results for integer values of $m$, but the question demands that $m>0$ be any real number.

I also tried changing the functions to $\Gamma$ ones, but that didn't lead anywhere meaningful.

Does anyone know how to approach this?

P.S. It seems to be an easy question: the marks to be awarded against it is unity in modulus.

Answer 1

$$B(m,1/2)=\int_0^1(1-x)^{m-1}x^{-1/2}dx\text{ (from the definition)}\\ =\int_{-1}^1(1-x^2)^{m-1}dx\\ =2^{-1}\int_0^14^m(y-y^2)^{m-1}dx$$ The substitution used in the last equation is $x=2y-1$.
Can you finish it by using the definition of Beta function?
Edit: The substitution used in the second equation is $x\mapsto x^2$, then use the fact that the integrand is an even function.

Answer 2

For any value of $m$ $$\frac{{B}(m, \frac{1}{2})}{{B}(m, m)}=2^{2m-1}$$

If you use the gamma function $$B(m,n)=\frac{\Gamma (m) \Gamma (n)}{\Gamma (m+n)}$$ $$\frac{{B}(m, \frac{1}{2})}{{B}(m, m)}=\frac {\sqrt{\pi }\frac{ \Gamma (m)}{\Gamma \left(m+\frac{1}{2}\right)} } {\frac{\Gamma (m)^2}{\Gamma (2 m)} }=\sqrt{\pi }\frac{ \Gamma (2 m)}{\Gamma (m) \Gamma \left(m+\frac{1}{2}\right)}$$Use Stirling approximation up to any order of your choice, continue with Taylor series to prove it.