I have this specific linear non-constant coefficient, non-homogenous recurrence relation that I'm trying to solve but stuck with. I'll show the recurrence and steps I've done so far.
Recurrence: $\displaystyle h_{n+1} = \dfrac{n}{n+1} h_{n}+\dfrac{p_{S}}{n+1}$, where $p_{S}$ is a constant between $0$ and $1$. And $n\geq0$, so we assume $h_{0}$ has been given.
I will show the steps I've done so far.
Let $\displaystyle H(z) = \sum_{n=0}^\infty h_{n}z^n$. Multiplying both sides by $z^n$ and summing from $0$ to $\infty$, we get
$$\sum_{n=0}^\infty h_{n+1}z^n = \sum_{n=0}^\infty \frac{n}{n+1}h_{n}z^n +p_{S}\sum_{n=0}^\infty\frac{z^n}{n+1}.$$
The LHS is $\dfrac{1}{z}(H(z)-h_{0})$. The second term of the RHS is $\displaystyle\dfrac {p_{S}}{z}\int H(z)dz$.
But I've tried for half an hour to figure out how to write the first term of the RHS in terms of derivatives or integrals of $H(z)$ and I have no idea how to progress. If anyone can provide hints so I can proceed on my own, I'll gladly take it. Thanks so much!
P.S. if I have an integral equation as above of $H(z)$, is it valid to take derivatives to turn it into a differential equation instead?