In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them.

• What are these techniques?

These similarities allow one to construct umbral proofs, which, on the surface cannot be correct, but seem to work anyway.

• What does "seem to work" mean here?
• It seems that umbral calculus is a mathematical idea with almost no uses, why? (At least it's not so famous as calculus and algebra, for example.)
• Calculus and algebra are massive branches of mathematics encompassing thousands of techniques each. It is far to much to demand of umbral calculus, which is essentially a single technique, to be equally useful. Sep 6, 2012 at 3:50
• "What are these techniques?" Aren't several examples given in the very Wikipedia article you're quoting?
– user856
Sep 6, 2012 at 4:01
• Roman's Advanced Linear Algebra has a nice chapter on this, that might be a good place to read. Sep 6, 2012 at 4:02
• The Wikipedia article itself already gives good examples and also has references. Sep 6, 2012 at 4:14
• You can download some of Roman's articles on umbral calculus here: romanpress.com/MathArticles/MathArticles.htm
– wj32
Sep 6, 2012 at 6:06

Umbral relations shadow those of the basic binomial transform, revealing underlying connections between diverse areas of math (as Leibnitz himself predicted--see H. Davis "Theory of Linear Operators"):

[Edit June 17, 2021: A slightly different introductory presentatation is at MO here.]

I) Umbral notation is brief and suggestive (courtesy of Blissard and contemporaries):

$$\displaystyle (a.)^n= a_n \; \; \;$$ (umbral variable and lowering of superscript).

Expressing binomial convolution simply:

$$\displaystyle (a. + b.)^{n} = \sum_{k=0}^{n} \binom{n}{k} a_{k} b_{n-k} \; \;$$ (be careful to evaluate $$(a.+b.)^0=a_0b_0$$ and $$(a.+b.)^1=a_0b_1+a_1b_0$$), $$\displaystyle e^{a. \; x}= \sum_{n \ge 0} a_n \frac{x^n}{n!} \; \; ,$$

$$\displaystyle e^{a.\;x}\; e^{b.\; x} = e^{(a. + b.)x}\; \; .$$

A more precise notation is to use $$\langle a.^n \rangle = a_n$$ to clearly specify when the lowering op, or evaluation of an umbral quantity, is to be done. E.g.,

$$\langle a.^n a.^m\rangle=\langle a.^{n+m}\rangle= a_{n+m} \ne a_n a_m= \langle a.^n\rangle\langle a.^m\rangle$$

and $$\langle\exp[\ln(1+a.x)]\rangle=\langle(1+a.x)\rangle=1+a_1x$$

$$\ne \exp[\langle\ln(1+a.x)\rangle]=\exp \left[ \sum_{n \ge 1} \langle\frac{a.^nx^n}{n}\rangle\right]=\exp\left[\sum_{n \ge 1} \frac{a_nx^n}{n}\right]\; .$$

II) Same for umbralized ops, allowing succint specification and derivation of many relations, especially among special functions. A good deal of umbral calculus is about defining these ops for special sequences, such as the falling $$(x)_{n}=x!/(x-n)!$$ and rising factorials $$(x)_{\bar{n}}=(x+n-1)!/(x-1)!$$ and Bell polynomials $$\phi_n(x)$$.

Examples:

$$(:AB:)^n = A^n B ^n$$  (defn. for order preserving exponentiation for any operators )

$$(xD)^n = (\phi.(:xD:))^n = \phi_n(:xD:) \; \; ,$$

$$e^{txD} = e^{t \phi.(:xD:)} \; \; .$$

From $$xD \; x^{n} = n \; x^{n}$$, it's easy to derive

$$e^{t\phi.(x)} = e^{x (e^t-1)} \; \; .$$

(See this MO-Q for the o.g.f.)

III) Umbral compositional inverse pairs allow for easy derivations of combinatorial identities and reveal associations among different reps of operator calculi.

Look at how this connects the distributive operator exponentiation $$:xD:^n=x^nD^n$$ to umbral lowering of superscripts. The falling factorials and Bell polynomials are an umbral inverse pair, i.e., $$\phi_n((x).)=x^n=(\phi.(x))_n$$. This is reflected in the functions $$\log(1+t)$$ and $$e^t-1$$, defining their e.g.f.s $$e^{x\log(1+t)}$$ and $$e^{x(e^t-1)}$$, being regular compositional inverses and to the lower triangular matrices containing the coefficients for the polynomials (the Stirling numbers of the first and second kinds) being multiplicative inverses, so we can move among many reps to find and relate many formulas. For the derivative op rep,

$$((xD).)^n=(xD)_n=x^nD^n=:xD:^n=(\phi.(:xD:)).^n=(\phi.(:xD:))_n,$$

so we have a connection to the umbral lowering of indices

$$:xD:^n=((xD).)^n=(xD)_n=x^nD^n.$$

IV) The generalized Taylor series or shift operator is at the heart of umbral calculus:

$$e^{p.(x)D_y}f(y) = f(p.(x)+y) \; ,$$

(e.g., this entry on A class of differential operators and another on the Bernoulli polynomials) with special cases

$$e^{:p.(x) D_x:} f(x) = f(p.(x) + x) \; ,$$ and

$$e^{-(1-q.(x))D_y}y^{s-1} \; |_{y=1} = (1-(1-q.(x)))^{s-1} \; ,$$ giving a Gauss-Newton interpolation of $$q_n(x)$$ (shadows of the binomial relations).

It can often be used to easily reveal interesting combinatorial relations among operators. A simple example:

$$e^{txD} f(x) = e^{t\phi.(:xD:)} f(x) = e^{(e^t-1):xD:} f(x) = f(e^{t}x) \; .$$ You could even umbralize $$t$$ to obtain the Faa di Bruno formula. Try discovering some op relations with the Laguerre polynomials (hint--look at $$:Dx:^n= D^nx^n$$. Cf. Diff ops and confluent hypergeometric fcts.).

As another example (added May 2015) of the interplay between differential operators, umbral calculus, and finite differences, note the relations for the Bell polynomials

$$\phi_{n}(:xD_x:)= \sum_{k=0}^n S(n,k)x^kD_x^k = (xD_x)^n=\sum_{j=0}^\infty j^n \frac{x^jD^j_{x=0}}{j!}=\sum_{j=0}^\infty (-1)^j \left[\sum_{k=0}^j(-1)^k \binom{j}{k}k^n\right] \frac{x^jD_x^j}{j!} \;$$

and apply these operators on $$x^m$$, $$e^x$$, and $$x^s$$. (The $$S(n,k)$$ are the Stirling numbers of the second kind.)

I've used the power monomials $$x^n$$ and their associated raising and lowering ops, $$x$$ and $$D_x$$, but these relations are shadowed by the raising and lowering ops of all umbral sequences $$p_n(x)$$ such that $$R \; p_n(x) = p_{n+1}(x)$$ and $$L \; p_n(x) = n \; p_{n-1}(x)$$. (Shadows of Lie and quantum mechanics here also.)

V) (Added Sept. 2020): The generalized Chu-Vandermonde identitiy for the discrete convolution of binomial coefficients--integral to understanding properties of confluent hypergeomeric functions and their diff op reps--is easily derived from the umbral Sheffer calculus.

Binomial Sheffer sequence of polynomials (BSP) have e.g.f.s of the form

$$e^{x \; h(t)} = e^{t \; B.(x)},$$

where $$h(t)$$ and is invertible and vanishes at the origin. This implies

$$e^{(x+y)h(t)} = e^{t \; B.(x+y)} = e^{xh(t)}e^{yh(t)} = e^{t \; B.(x)}e^{t B.(y)} = e^{t(B.(x)+B.(y))},$$ so follows the accumulation property $$(B.(x)+B.(y))^n = B_n(x+y).$$ The Stirling polynomials of the first kind, $$ST1_n(x) = (x)_n$$, are a BSP with $$h(t)=\ln(1+t)$$ and

$$\binom{x}{k} = \frac{ST1_k(x)}{k!},$$ so $$(ST1.(x)+ST1.(y))^n = ST1_n(x+y)$$

implies directly the Chu-Vandermonde identity

$$\binom{x+y}{n} = \sum_{k=0}^n \binom{x}{k} \binom{y}{n-k}.$$

• I'm having introductory lectures in combinatorics. I'm amazed that Umbral Calculus is a lot similar to some of the things I've seen there, I thought it was nearer to real analysis. Although, as you pointed in the start of the text: It might have a lot of connections. Jan 2, 2015 at 12:19
• Master basic combinatorics first (binomials, Stirling numbers, Chu-Vandermonde identities--keeping an eye on their apps in analysis--check OEIS entries also) and linear algebra (linear functionals, dual spaces). Then you should be ready for Rota, Roman, et al., as Michael suggests. Jan 2, 2015 at 19:29
• Can't wait? Start studying the Appell polynomials, in particular, the Hermite and Bernoulli polynomials, and the three binomial Sheffer sequences the rising and falling factorials and Bell / Touchard polynomials. In parallel, look at finite differences and their differential operator reps. (Wiki refs again). I think after that you can easily make your own choices about what relevant operator calculi and integral transforms to explore. Use the umbral notation to simplify expressions and suggest formulas. Jan 2, 2015 at 19:50
• Yep, the intersections inform. Also, you can often contribute something new yourself. Look at noncrossing partitions A134264. Jan 3, 2015 at 10:32
• @Neil: Finding the zeros of polynomials is not related to umbral calculus. Apr 7, 2015 at 1:57

From the Wikipedia article: "The combinatorialist John Riordan in his book Combinatorial Identities published in the 1960s, used techniques of this sort extensively." There you see the classical umbral calculus. Basically he pretends subscripts are exponents, and somehow it works. Take a look at that book. It doesn't require a lot of apparatus.

The 1978 paper by Roman and Rota, cited in the article, is the beginning of a technique for making the classical umbral calculus rigorous.

The 1975 paper by Rota, Kahaner, and Odlyzko appears to be a paper about Sheffer sequences, which are certain sequences of polynomials (see the Wikipedia article titled "Sheffer sequence"). If you lay that paper and the 1978 paper side-by-side, you can see that they're really two different ways of looking at the same thing.

In the mean time, look at the concrete examples in the Wikipedia article that you cited.

Is it useful? I think one could argue about that. But I don't want to try to make the case for its utility in research.

• What you mean with useful? I've read a few stuff on the differences of pure and applied mathematics, considering the view of pure mathematics, this definition seems to don't exist. Sep 6, 2012 at 6:18
• For a simple use of umbral substitution in a current research paper, see p. 2 of "KP hierarchy of Hodge Integrals" arxiv.org/abs/0809.3263. Sep 16, 2016 at 18:50
• That is, pages 2, 4 and 7. Sep 16, 2016 at 21:19
• A nice example of the use of UC in contemporary research: "On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops" Miguel A. Méndez and Rafael Sánchez arxiv.org/abs/1707.00336 Aug 25, 2021 at 14:03