## Task 6B - 221246B

In the investigation of frequency distributions in mathematical statistics, functions appear that are defined for finitely many natural numbers and are required to satisfy so-called functional equations (equations between different function values).

An example of this is the following:

Given a natural number $n \geq 2$ and a real number $p$ with $0<p<1$.

Determine (in dependence on $n$ and $p$) those functions $f$ with the set $\{1,2, \ldots, n\}$ as the domain, which for $k=1,2, \ldots, n$ satisfy the following equation (1):

$$
\sum_{x=0}^{n} x \cdot(x-1) \cdot(x-2) \cdot \ldots \cdot(x-k+1) \cdot f(x)=n \cdot(n-1) \cdot(n-2) \cdot \ldots \cdot(n-k+1) \cdot p^{k}
$$

Hint: For $k=1$, equation (1) is to be interpreted as

$$
\sum_{x=0}^{n} x \cdot f(x)=n \cdot p
$$