# finite calculus

(“ to the falling”) is the falling (factorial) power :
e.g. = and

The inverse of is the anti-derivative (integration) operator
, the indefinite integral of , is the class of functions whose derivative is .
The “” for indefinite integrals is an arbitrary constant.

The inverse of is the anti-difference (summation) operator
, the indefinite sum of , is the class of functions whose difference is .
The “” for indefinite sums is any function such that .

Fundamental theorem of the sum calculus:

Leibniz’s rule for the th derivative of the product of two functions and :

Leibniz’s rule for differences:

### falling powers

When , since , we get:

Since :

“factorial binomial theorem”

Like
And similarly for each and .

While

The coefficients are the Stirling numbers of the first kind.

The coefficients are the Stirling numbers of the second kind.

Knuth et al – Concrete Mathematics