Simplify Γ(2α) in Γ(α) terms

Question

I need to simplify Γ(2α) in Γ(α) terms where α is a real number greater than 1. (I need to cancel out Γ( . ) terms in a simplification). Shall accept a product term with Γ(α), like (2α - 1)*...αΓ(α). Thanks in advance.

Answer 1

Take the product $$ (2\alpha-1)\cdots\alpha\Gamma(\alpha) $$ and work your way from the right. The two rightmost terms are $\alpha\Gamma(\alpha)$. By the defining property of the $\Gamma$ function, this is equal to $\Gamma(\alpha+1)$.

Write down the total product after this simplification. What are the two rightmost terms? Can that be simplified? What is the resulting total product? Can that be simplified? How far can you keep going? What's the end result?

PS. This does, of course, assume that $\alpha$ is a positive integer.