I've always been interested in situations where we can apply "illegal" operations to objects and still solve problems (as seen here, say), and a common justification for these techniques is the umbral calculus. There are already many questions on this site about what the umbral calulus is (see here and here), and many applications where people solve problems with it (see here, and here).

I've flipped through Roman's classic The Umbral Calculus, and while I enjoyed the exposition of the method itself, I was left with an important question:

How can I tell when a problem is amenable to the umbral calculus?

I feel like I'm still unclear on this because of the lack of exercises in Roman's text. Ideally, there would be a book like generatingfunctionology for the umbral calculus, but I realize this is a high bar to clear.

Does anyone have any references which do contain exercises and intuition for which problems the umbral calculus can solve?

Additionally, does anyone have intuition that they themselves are willing to share regarding when we should think to reach for this tool?

NB: I originally asked this in a comment on this answer, because I wasn't sure if it warranted a full question (I admit it's quite soft), however I was encouraged to ask it as a separate question. Hopefully this will give more people an opportunity to chime in ^_^

Thanks in advance!



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