$$F(z) = \sum_{n=0}^\infty a_n x^n $$(adsbygoogle = window.adsbygoogle || []).push({});

$$\partial_zF(z) = \sum_{n=0}^\infty (n+1)a_{n+1}x^n $$

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.

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# Finite Difference Expressed As a Probability Generating Function

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