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Umbral Calculus

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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)2Bn = 0. After binomial theorem expansion, the Bk is replaced with the Bk to get a recursive formula for the Bernoulli numbers [2]:
umbral notation for bernoulli numbers

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
Bxn = Bn.
The “lowering of the index” uses the relationship (X – 1)n = Xn and adding B to both sides to get B(X – 1)n = B(Xn).

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:
umbral calculus example


[1] Bucchianico, A. (1998). An introduction to Umbral Calculus. Retrieved May 4, 2021 from:
[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:
[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

Stephanie Glen. "Umbral Calculus" From Calculus for the rest of us!

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