Tried reading wikipedia, watched some videos introducing it. Still have no idea what it’s about.
In: Mathematics
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!
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