Tried reading wikipedia, watched some videos introducing it. Still have no idea what it’s about.
In: Mathematics
I’ve only just read the wikipedia article and have not really deep-dived into videos or references, but…
It looks like “doing Umbral Calculus” is just using some “generatingfunctionology”-like combinatorial techniques/methods on some certain complicated power series to convert them into other forms more amenable to solving the calculus problem at hand.
Basically, you have a calculus problem involving a particular kind of power series that is hard/tedious to “do calculus” with, your combinatorically-minded colleague says *”Hey, that kinda looks like a Blah-Blah-Blah generating function!”*, and when you look up what “Blah-Blah-Blah generating function” means in combinatorics you find some combinatorial identities pertaining to it, you find one such identity that looks remarkably convenient for your purposes if only it worked… so you decide to as-if “translate” your calculus series problem into it’s Blah-Blah-Blah version, then use the convenient identity to get it into a different form, then as-if “translate” that convenient identity back to it’s original-problem calculus context… and somehow it seems to work out!?
It feels *shadowy* and *mysterious* because the “translate” steps at either end feel very “loosey goosey” and non-rigorous, but the resulting calculus-problem identity can be proven valid (with time and effort) by other means.
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