I need some hints or keywords for the following problem: I am looking for a family $f_{k,n}(z)$ of functions ($k,n \in \mathbb{N}$) in $z \in [0,1]$, which satisfy the following functional/differential equation: $$ f_{k,n}(z) = \frac{n}{k} \cdot z \cdot f^{'}_{k+1,n}(z) \,,\qquad (z \in (0,1)) $$ with $f_{k,n}(0)=0$ and $f_{k,n}(1) = 1$ for all $k,n \in \mathbb{N}$.
Because of the shift k+1 in the parameter and the conditions at $0$ and $1$, I cant (or dont know) how to solve this.
Without the conditions at $z=0$ and $z=1$ I got a solution for $z \in [0,1)$, but these conditions are really important for my problem.
Does anyone know any good keywords I should search for to solve such equations?
Thanks in advance.