Introduction to Umbral Calculus

Definition: Discrete Derivative (or Forward Difference Operator)

Given , define

which is “as infinitesimal” as the domain can get.

Examples

which follow like a Pascal’s triangle. Let () be the coefficient multiplying the -th power of the discrete derivative of . Then:

Then (for )

Note that it’s the same relation as in the Pascal triangle¹, for which

Definition: Falling Factorial

For , define

Note that

Examples

Theorem 1: Discrete derivative for falling factorials = Calculus for natural powers

For , we have

Proof:

In a more elucidative form:

(Third line is Pascal’s Triangle’s relationship)

Derivative of Exponential Functions

Given , we have (let )

Note that it follows that

Definition: Discrete Antiderivative

Given , we say that is the antiderivative of if

Denote the antiderivative operator as

Corollaries: Antiderivative of falling factorials and real numbers

Example: Fibonacci numbers!

.

Observe that

and so it acts as a “step-down” operator.

Likewise, the discrete antiderivative acts as a “step-up” operator, since it yields the function that, when derived, results in (i.e. ):

Theorem 2: Discrete Fundamental Theorem of Calculus

Given , where is the antiderivative of . Then

Proof: It’s essentially a telescopic sum. Note that the same proof sketch appears in the Fundamental Theorem of Calculus, and even in Stokes’ Theorem.

Note that it’s essentially the same as the fundamental theorem of calculus, except for the , which is analogous to an infinitesimal : all natural numbers are at least 1 unit apart from each other, which doesn’t really happen in the real numbers (nor in the rationals), since it’s a dense set.

Product Rule for Discrete Derivative

Proof:

Note that the term appears here, and not in regular Calculus, since there is a in place. In Discrete Calculus, we can only go as close as +1 from any integer, so these terms that would vanish in Standard Calculus don’t vanish here.

Summation by Parts

Proof:

Note that it’s very similar to its analogue in Differential Calculus, in which

Applications: Evaluating (some) series!

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Referências

• https://www.youtube.com/watch?v=FNh-Xkfnu1M

Footnotes

1. Check Wikipedia’s article on Pascal’s Triangle. ↩