(“ 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 .

Add together , to get:

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