Define, for

Then proof for all

And also prove for

I once asked someone this question and they said it looks like the Beta Function, thus maybe that will help.

If you are curious how I came to such a conjecture is because I was trying to describe a mathematical formula for the Normal Distribution and these two:

and

where is the probability.

Graph curves which look like the Normal Distribution, furthermore the second one is a density curve this is why its Area is one.