Another way of writing them is to use an underline:
Which is easier when writing, and also clearer. (This isn't possible
with the current math font)
In general a factorial polynomial of degree n, (yk
or kn) is: [1.01]
We assume that n is an integer greater than zero (A natural number).
We can call this k to the n falling (because there is a rising
version!) with step h.
k to the n+1 falling is:
Which, simplifying the last term: [1.02] k(0) is defined as 1
Finding
the First Difference
By definition, the first difference for the factorial polynomial, k(n), is [1.03]
Substituting our values from 1.01 and 1.02 for kn+1
and kn in 1.03: [1.04]
Factorising gives us: [1.05]
And further simplifying the final term by cancelling the x's and
rounding up the h's: [1.06]
We note that, substituting n-1 for n in 1.06:
Simplifying the final factor: [1.07]
First Difference and General Formula for n>0
From 1.06 substituting 1.07, we have:
[1.08]
So we can determine any of the differences using 1.08, for instance:
In general, the mth difference is: [1.09]
This is reminiscent of differentiating using the infinitesimal
calculus. [1.10]
1.08 also reminds us of similar
result for regular polynomials, repeated below:
With
regular polynomials, the difference isn't so neat as that with
factorial polynomials. However, we can convert regular polynomials to
factorials and obtain clearer results for differences.
Often, the factorial polynomials we use have a step of 1, or h=1, so: k(n)=k(k-1)(k-2)...(k-n)(k-n+1)
[1.11]
And the mth difference when h=1 is:
[1.01]
[1.02]
[1.03]
[1.04]
[1.05]
[1.06]
[1.07]
[1.08]
[1.09]
[1.10]
■[1.12]