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. –  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