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 .
The name Umbral Calculus was invented by Sylvester, “that great inventor of unsuccessful terminology” . 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 – Bn = 0. After binomial theorem expansion, the Bk is replaced with the Bk to get a recursive formula for the Bernoulli numbers :
The reason why lowering the index “works” has its roots in expressing an infinite sequence of numbers by a transform . 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 . For example, umbral calculus has been used to solve martingale problems  and recurrences as well as counting lattice paths .
Despite its simplicity, the early development of umbral calculus was not without its problems. For example, the following “rule” is what Roman & Rota  call “baffling” as seemed to imply that a + a ≠ 2:
 Bucchianico, A. (1998). An introduction to Umbral Calculus. Retrieved May 4, 2021 from: https://www.researchgate.net/publication/2471188_An_introduction_to_Umbral_Calculus
 Roman, S. and G.-C. Rota (1978). The umbral calculus. Adv. Math. 27, 95–188.
 Ray, N. Universal Constructions in Umbral Calculus. Retrieved May 4, 2021 from: http://www.ma.man.ac.uk/~nige/ucuc.pdfH.
 Hammouch, H. (2004). Umbral Calculus, Martingales, and Associated Polynomials. Stochastic Analysis and Applications
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. 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 CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/umbral-calculus/
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