**Umbral calculus** (also called *Blissard Calculus* or *Symbolic Calculus*) is a modern way to do algebra on polynomials. It is a set of exploratory “rules” or a proof technique where indices of polynomial sequences are treated as exponents; Generally speaking, it’s a way to discover and prove combinatorial identities, but it can also be viewed as a theory of polynomials that count combinatorial objects [1].

The name *Umbral Calculus* was invented by Sylvester, “that great inventor of unsuccessful terminology” [2]. The calculus is based around an *umbra*, symbol *B*, which comes from the Latin *umbral*. Although it is a “shady” way to approach problems, it actually works!

## Umbral Calculus Derivation of Bernoulli Numbers

A well known example of umbral notation is the representation of Bernoulli numbers by

(*B* + 1)^{2} – *B*^{n} = 0. After binomial theorem expansion, the B^{k} is replaced with the B_{k} to get a recursive formula for the Bernoulli numbers [2]:

The reason why lowering the index “works” has its roots in expressing an infinite sequence of numbers by a transform [2]. In other words, a linear transform B can be defined as

*Bx*^{n} = *B*_{n}.

The “lowering of the index” uses the relationship (*X* – 1)^{n} = *X*^{n} and adding B to both sides to get *B*(*X* – 1)^{n} = *B*(*X*^{n}).

## Development of Umbral Calculus

Umbral calculus is becoming more well known as it heads towards maturity, with applications in several mathematical areas [3]. For example, umbral calculus has been used to solve martingale problems [4] and recurrences as well as counting lattice paths [5].

Despite its simplicity, the early development of umbral calculus was not without its problems. For example, the following “rule” is what Roman & Rota [3] call “baffling” as seemed to imply that a + a ≠ 2:

## References

[1] Bucchianico, A. (1998). An introduction to Umbral Calculus. Retrieved May 4, 2021 from: https://www.researchgate.net/publication/2471188_An_introduction_to_Umbral_Calculus

[2] Roman, S. and G.-C. Rota (1978). The umbral calculus. Adv. Math. 27, 95–188.

[3] Ray, N. Universal Constructions in Umbral Calculus. Retrieved May 4, 2021 from: http://www.ma.man.ac.uk/~nige/ucuc.pdfH.

[4] Hammouch, H. (2004). Umbral Calculus, Martingales, and Associated Polynomials. Stochastic Analysis and Applications

Volume 22, Issue 2. pp 443-447.

[5]. Humphreys, K. & Niederhausen, H. (2004). Counting lattice paths taking steps in infinitely many directions under special access restrictions. Theoretical Computer Science 319, 385 – 409

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