How to evaluate: $\int_{0}^{1-z} y^{j-1} (1-z-y)^{n-k} dy$

Question

How can I compute the integral:

$$\int_{0}^{1-z} y^{j-1} (1-z-y)^{n-k} dy\quad\text{where}\ z \in (0,1) $$

Had it not been for $z$ , the integral would look like an incomplete beta function but what about now?Any suggestions as to how to proceed?

Thank you.

Answer 1

Do a change of variables, replacing $y$ by $(1-z)y$. Then it is literally a beta integral, with a leading factor a power of $(1-z)$.