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.

In layman’s terms, the “original” theory of “umbral calculus” (called that because it seemed to involve shadowy operations, but produced results verifiable with cumbersome but provable methods) was a seemingly paradoxical relationship between subscript index numbers (e.g. “*Bₙ*(x)”) in one type of equation, and superscript exponent numbers in related equations (e.g. y^n + y^k-n or the like). This seemed *very weird* to the mathematicians who were studying this, because normally there is either no special relationship at all between index numbers and exponents, *or* the relationship is a trivial one, that comes from just flat defining something to be that way. Early on, this did not seem to be what was happening. It seemed like an almost magical connection that didn’t need to be defined.

Now (or, rather, 50ish years ago), we can see that there *is* a specific operation happening here which makes the “magic” happen. It’s called a “linear functional,” which (for this specific application) is a way of mapping exponential things to indexed numbers so that the map is easier to work with, but one that preserves all of the important structure and behavior of the original exponents. This general *idea,* of converting to a different perspective where a problem becomes (in some sense) “easy”, solving it, and then converting back to the original form, is extremely common in math and physics. I recently (in another ELI5) mentioned the “method of images,” a technique used for solving differential equations, e.g. “what does the electric field around this conductor look like with these charges near it?”, by effectively pretending that there are extra “mirror” charges on the other side, which let you exploit symmetry arguments and the like. Or, to use a different example, the proof of Fermat’s last theorem came from proving something seemingly unrelated (an equivalence relationship between equations like a^n + b^n = c^n and a different branch of mathematics), and then showing that a then-recently proven conjecture in that other area of math necessarily meant that Fermat’s last theorem had to be true.

This is very esoterical even by mathematics standards and is going to be hard to ELI5, but I’m giving it a try:

Lets look at three things that at first seem pretty unrelated:

– **Monomials x^^n = x·x·x·…·x** as basic expressions.

– **Differentiation D(f) = f'(x)**, the slope of a function f(x), the momentary change of f.

– **Binomial Theorem**: (x+y)^^n = 1· x^^n + n· x^^n-1 y + … + (n;k)· x^^n-k y^^k + … + n· x y^^n-1 + 1· y^^n where (n;k) = n!/(k!·(n-1)!) are the _binomial coefficients_.

Those three things have some relationships. We can observe that the Binomial theorem involves a ton of monomials and that a monomial’s derivative D(x^^n) = n·x^^n-1 is particularly simple and again a monomial. We can even get binomial coefficients from differentiation: differentiating x^^n not just once but k times gives us D^k( x^n ) = n·(n-1)·(n-2)·…·(n-k+1) · x^^n-k = k! · (n;k) · x^^n-k .

Lets look at something similar yet different:

– **Falling factorials x^^<n = x·(x-1)·(x-2)·…·(x-n+1)** as expressions in x.

– **Finite (upper) difference ∇(f) = f(x+1)-f(x)**, the change of f from now to 1 later.

– The **Vandermonde Identity**: (x+y)^^<n = 1· x^^<n + n· x^^<n-1 y + … + (n;k)· x^^<n-k y^^<k + … + n· x y^^<n-1 + 1· y^^<n

We can check that those actually satisfy again the same relationships: Vandermonde’s Identity looks exactly like the Binomial Theorem, but with (…)^^<stuff instead of (…)^^stuff . The finite difference of a falling factorial is quickly calculated to be ∇( x^^<n ) = n·x^^<n-1 . Similar for many other relationships.

That is what constitutes an **Umbra** (Latin: shadow): it “behaves like monomials and differentiation” in the sense of sharing certain common formulas such as the **binomial-type rule** and a pseudo-differentiation (“delta operator”) which lowers exponents (in two meanings, even!). Each time we get the corresponding formula by replacing x^^k by the corresponding k-th polynomial.

The three aspects we have now seen twice are so deeply interconnected that it essentially suffices to have any one of them to automatically get the others.

But even better, **one can actually create any umbra from any other** by what is almost magic! Shadow magic, one could say. Lets look at an example:

Lets act slightly insane and wonder what something like e^^D should _be_? Yes, that’s e to the power of differentiation!

One might have seen that e^^x = 1 + x + x²/2 + x³/6 + … + x^^k/k! + …, the _exponential (Taylor) series_. So e^^D might be 1 + D + D²/2 + D³/6 + … whatever that now means… and indeed, if we interpret D^k as k-fold differentiation and throw a function f in then we get a new function (e^^D)(f) = f + f'(x) + f”(x)/2 + f”'(x)/6 + …, which looks like it might actually make sense (we interpreted the initial 1 not as the number 1 but as D^^0 , taking the 0-th differential, so not doing anything to the function).

Hence e^D is not a number, it instead takes a function and outputs another function. But what new function is this? If one checks carefully or knows Taylor series it turns out that ( e^^D )(f) = f(x+1). Simpler than one might think at first. And thus we can finally relate two umbrae we saw in a single formula:

**∇ = e^^D – 1**, or by “solving for D” also **D = log(1+∇)**.

This kind of insanity is a full machine that produces as many umbrae as we could ever wish for: just pick any umbra δ you know such as the one associated with D, then write down any (possibly infinite) sequence of the form S(δ) = δ + a·δ² + b·δ³ + … . Throw in any function f and call the result for example Д(f).

If you want to recover the polynomials, you then recursively search for pseudo-antiderivatives: P[0] = 1, Д(P[n]) = n·P[n-1] and such that it evaluates to 0 at x=0.

Time for some examples, old and new, all of them somewhat famous sequences of polynomials and how they arise from some Д:

– **Powers** x^^n from differentiation D.

– **Falling factorials** x^^<n = x·(x-1)·(x-2)·…·(x-n+1) from (upper) finite differences ∇(f) = f(x+1)-f(x). We have ∇ = e^^D – 1.

– **Rising factorials** x^^>n = x·(x+1)·(x+2)·…·(x+n-1) from (lower) finite differences Δ(f) = f(x)-f(x-1). We have Δ = 1 – e^^-D .

– **Central factorials** x·(x-1+n/2)^^<n-1 = x·(x+1-n/2)^^>n-1 , so the polynomials 1, x, x², (x+½)x(x-½), (x+1)x²(x-1), (x+3/2)(x+½)x(x-½)(x-3/2), (x+2)(x+1)x²(x-1)(x-2), … from (central) finite differences ◇(f) = f(x+½) – f(x-½). This time we have ◇ = e^^D/2 – e^^-D/2 .

– **Abel polynomials** x·(x-an)^^n-1 for any fixed a from Д = D·e^a·D .

– **Bell polynomials** from log(1+D).

– any many many more including Bernoulli, Cauchy, Mittag-Leffler, …

And each single one of them satisfies their instance of the Binomial Theorem!

I think if I interpret the explanations correct, a really simple way to put it is like:

Some math nerds were looking at some really hard formulas, and they thought, “Man, if THIS math worked like this OTHER kind of math that looks a lot like it, things would be a lot easier.” But if they tried to logically prove the two things should work alike, there was no way to form an argument that they should.

Then some smart-aleck said, “You know what, what if I do it anyway?” and it turned out *it worked*. There’s still not a great logical framework for explaining why. But there’s also no proof *it doesn’t* despite many people trying.

So it’s basically a big math shortcut, like noticing that if you add up all the digits of a number, and that sum is a multiple of 3, then that number is a multiple of 3. Try it. 12345’s digits sum to 15, and 15 is divisible by 3. Ta da, the number is divisible by 3.

That’s kind of crazy, because our number system wasn’t designed to MAKE that true. And there’s probably not some actual rule that defines WHY it happens. It’s just some kind of weird coincidence that is ALWAYS true. (Maybe there is a proof, and I’m unaware.)

My feeling is Umbral Calculus is a lot like that. It’s a weird trick that lets mathematicians treat one kind of complicated equation as if it were a different kind. Sometimes that makes using the equation for what they want simpler. If you are very strict about logic and proofs it’s *not supposed to work*. But it does.

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