I am trying to find a general expression from a recursion.
Here it goes: $(x+i)P_i = (i+1)P_{i+1} + \frac{x}{2} P_{i-1}$
$i$ goes from $0$ to $S$.
How can I calculate a generic $P_i$ in terms of $P_0$? So, if $P_i = C \times P_0$, what is $P_0$?
Is it possible to solve this in Mathematica? What is the function I'm looking for?
You can assume $P_0 = 1$.
Thanks.
[UPDATE]
I am not quite sure if this would help. I started from $P_0$ and have the following equations: $P_{1}=P_{0}x$
$P_{2}=P_{0}(\frac{1}{2!})(x^2+\frac{x}{2})$
$P_{3}=P_{0}(\frac{1}{3!})(x^3+\frac{3x^2}{2}+x)$
$P_{4}=P_{0}(\frac{1}{4!})(x^4+3x^3+\frac{19x^2}{4}+3x)$
$P_{5}=P_{0}(\frac{1}{5!})(x^5+5x^4+\frac{55x^3}{4}+20x^2+12x)$
I tried to find polynomials that will represent above, but I could not find one. These are basically balance equations for a two dimensional markov chain. I solved it numerically, but I want to have a analytical solution as well.