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.)
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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. –  Alex Becker Sep 6 '12 at 3:50
"What are these techniques?" Aren't several examples given in the very Wikipedia article you're quoting? –  Rahul Sep 6 '12 at 4:01
Roman's Advanced Linear Algebra has a nice chapter on this, that might be a good place to read. –  James S. Cook Sep 6 '12 at 4:02
The Wikipedia article itself already gives good examples and also has references. –  Qiaochu Yuan Sep 6 '12 at 4:14
You can download some of Roman's articles on umbral calculus here: romanpress.com/MathArticles/MathArticles.htm –  wj32 Sep 6 '12 at 6:06

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 titles "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.

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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. –  Igäria Mnagarka Sep 6 '12 at 6:18