Question regarding ratio

Question

I was trying to prove some of the results related to beta function and I came across this step

Therefore, $$\frac{B(m+1,n)}{m}=\frac{B(m,n+1)}{n}=k,\quad\mathrm{say}\quad\ldots(i)$$ This yields $$k=\frac{B(m+1,n)+B(m,n+1)}{m+n}$$

Can anyone tell me how they got the value of $k$? Thanks in advance.

Answer 1

From the first equation, we get $km=B(m+1,n)$ and $kn=B(m,n+1)$. Thus, $$k=\frac{k(m+n)}{m+n}=\frac{km+kn}{m+n}=\frac{B(m+1,n)+B(m,n+1)}{m+n}$$

Answer 2

The first equation equates three different expresssion. We may pick any two to make an equation. Take the first and last, with a slight rearrangement:

$B(m+1,n) = m k$

then the second and last:

$B(m,n+1) = n k$

Now, just add these two equations I just wrote. The final step should be obvious.