(“ 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:
When , since , we get:
“factorial binomial theorem”
And similarly for each and .
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