Finite Difference Expressed As a Probability Generating Function

[MisterX]

$$F(z) = \sum_{n=0}^\infty a_n x^n $$
$$\partial_zF(z) = \sum_{n=0}^\infty (n+1)a_{n+1}x^n $$
So, we can begin to piece together some differential equations in terms of generating functions in order to satisfy some discrete recursion relation (which is the desired problem to solve). However I desire something more - perhaps a table of properties akin to what we might find for the eerily similar-yet-different z-transform.
$$a_{n+1} - a_n = g(n) \forall n \to M(F(z), F'(z), \dots) = 0 $$
Can we establish a property for the finite difference for example? What would be $M$ ? Intuitively we expect something like zF(z) + boundary terms.

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[Stephen Tashi]

This is like a inverse problem to the Frobenius method of solving differential equations. Maybe someone who has made up a lot of homework problems for that method would have insight.

We could begin by thinking about functions $F(z)$ that are polynomials of degree N instead of infinite series It might also be simpler to think about being given $a_n = G(n)$ and knowing $g(n)$ from the differences of $G(n)$. Thinking of polynomials as vectors in the usual way, the $a_i$ are the coefficients for basis vectors in the set $\{ 1, z, z^2, ..z^N\}$. The values of $G(n)$ define a vector. If we want a linear differential equation with constant coefficients we need to represent $G(n)$ as linear combination of the vectors in the set $S$ of vectors representing $\{ F(z), F'(z), F''(z),...F^N(z) \}$. If we want an DE whose coefficients are polynomials in $z$ then we add more vectors to S such as those representing $\{ zF'(z), z^2F'(z), ... zF''(z), z^2 F''(z)...\}$

Intuitively, a typical $G(n)$ defined by a vector of values randomly selected by someone creating a homework problem would give enough independent vectors in the set $S$ to include $G(n)$ in its span. But hastily making up homework problems is risky.!