I am looking for a solution to the following functional equation: \begin{align} (n+1) f(n+1)= (a n+b) f(n), n=0,1,... \end{align} where $a$ and $b$ are some positive constants. Moreover, $f(n)$ is positive and \begin{align} 0<f(n) &\le 1,\\ \sum_{n=0}^\infty f(n)&=1. \end{align}
I was able to find a solution if $a=b \le 1$ \begin{align} f(n)= \frac{c}{(1+c)^{n+1}}, \, c=\frac{1-a}{a}. \end{align}
My questions:
- Is there a systematic way to solve this equation?
- Is the solution to this equation unique?
Any reference would also be appreciated.