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Polynomial Factorials n Negative

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Negative Factorials

We define a factorial power thus:
1 [1.02]

Proof of Relationship Between Factorial Powers

We note:
k(n)=k(k-1)...(k-n+1) [1.03]

And by substituting (n+1) for n in 1.03, 
k(n+1)=k(k-1)...(k-n+1)(k-n) [1.04]

Therefore:
k(n+1)=k(n)(k-n)

Values for 0 and 1

Substituting 0 for n in 1.02 we find:
2 [1.05]
And substituring 1 for n:
3[1.06]

Expression for Negative Factorials

We have shown that k(0) is 1, and we make the following claim:
4[1.07]
By induction, we claim that if 1.07 is true then so is (substituting (-n+1 for -n)
5 [1.08]

We recall that
6 [1.09]
And
7 [1.10]

Hence, by substituting these in 1.07 and 1.08
8 [1.11]

9[1.12]
From 1.11 and 1.12 we note:
10
Which is true by definition (1.02). Hence because 1.07 is true for n=0, it is true for n=-1, and so true for all n.

Examples for n=-1, -2, -3


If n=(-1), using
k(-1)=1/(k+1)1
k(-1)=1/(k+1)

For n=(-2),
k(-2)=1/(k+2)2
k(-2)=1/(k+2)(k+1)

For n=(-3),
k(-3)=1/(k+3)3
k(-3)=1/(k+3)(k+2)(k+1)


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