I have a generating function that I'm trying to create. It has a general form to it:
$1 + cx + c(c-1)x^2 + c(c-1)(c-2)x^3 + \dots + c!x^c$
I would like to see a closed form for this function. I'd like something other than a summation. I'm particularly interested if someone could walk through how to derive the form for me. I was hoping that someone could carry on with what I've done and show me how to solve what's left.
My attempt
So I try to create a function $A(x)$ (which will be the closed form) using recurrences. Here I'm using techniques (and especially notation) from Wilf's Generatingfunctionology.
I set $a_0=1$. I then proceed to multiply by $(c-n)$. My guess is that the math should be something like the following:
$a_{n+1}=c \cdot a_n - \frac{d A(x)}{dx}$
I'm trying to say that the next term should be $c$ times the previous term minus $n$ times the previous term. The latter is represented by the derivative, I believe. So I guess that this could be correct. Unfortunately, I don't know for sure, and so I'm kind of lost at this point. I would greatly appreciate if someone could take it from here and explain the rest in detail.
The Differential Equation
Assuming that the last part is correct, the function becomes a differential equation:
$\frac{A(x)-1}{x}=c \cdot A(x) - \frac{d A(x)}{dx}$