Question of Relation between Beta and Gamma Function

$Relation between Beta and Gamma Functions formula , Beta and Gamma Functions$ ………(1)

Replacing m by (m+1) in equation no.1

$https://latex.codecogs.com/svg.image?B(m+1,n) = frac{Gamma (m+1) Gamma n}{Gamma(m+n+1 )}$ ………..(2)

Replacing n by (n+1) in equation no.2

$https://latex.codecogs.com/svg.image?B(m,n+1) = frac{Gamma (m) Gamma (n+1)}{Gamma(m+n+1 )}$ ………..(3)

Adding equation no.1 and 2,

$https://latex.codecogs.com/svg.image?B(m+1,n)+B(m,n+1) = frac{Gamma (m+1) Gamma n}{Gamma(m+n+1 )}+frac{Gamma (m) Gamma (n+1)}{Gamma(m+n+1 )}$

$https://latex.codecogs.com/svg.image? = frac{mGamma mGamma n+Gamma m.nGamma n}{Gamma (m+n+1)}$

$https://latex.codecogs.com/svg.image? = frac{Gamma mGamma n.(m+n)}{(m+n)Gamma (m+n)}$ [from recurrence formula Γ(n+1)=nΓ(n)]

$https://latex.codecogs.com/svg.image? = frac{Gamma( m)Gamma( n)}{Gamma (m+n)}$

= B(m,n)

$\therefore$ B(m,n) = B(m+1,n) + B(m,n+1) …….Proved that.

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