We can convert a polynomial to a factorial polynomial by dividing it by
k, k-1, etc.

k^{(0)} is defined as 1.

Example

As an example, we will convert x^{2} to factorials.
h is taken to be 1. We suppose:
x^{2}≡a_{0}k^{(2)}+a_{1}k^{(1)}+a_{2}

In general successively divide the polynomial by x, x-1,
x-2...(x-n+1).
For x^{2} we divide by x and x-1.

We can divide how we please, but Synthetic Division
is perhaps preferred.

Write out the table as below. The second line is for the coefficients:

x ^{2}

x ^{1}

x ^{0}

1

0

0

0

1

Divide by x:

x ^{2}

x ^{1}

x ^{0}

1

0

0

0

0

0

1

0

0

1

In the normal fashion for synthetic division, we write down the 0 (of
x-0!) and write down the coefficient of x^{2}, 1,
in the lower line.
We take 0x1 and write it below the coefficient of x, adding them.
Finally we take this sum, 0, multiply it by 0 and write it below the
coefficient of x^{0}.
We are now finished with the last column, the value is 0, so there is
no k^{(0)} term.

Next we divide by x-1:

x
^{2}

x
^{1}

x
^{0}

1

0

0

0

0

0

1

0

0

1

1

1

1

Following the procedure for synthetic division, we divide by x-1. The
resulting factorial form is:
x^{2}≡k^{(2)}+k^{(1)}

Next are the results for x^{3}.

x

x
^{3}

x
^{2}

x
^{1}

x
^{0}

1

0

0

0

0

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

2

2

1

3

x^{3}≡k^{(3)}+3k^{(2)}+k^{(1)}

We can convert any polynomial to factorial form using this method.
As a final example, we convert the following:
x^{5}+2x^{4}+3x^{3}+7x^{2}+5x+19

x

x
^{5}

x
^{4}

x
^{3}

x
^{2}

x
^{1}

x
^{0}

1

2

3

7

5

19

0

0

0

0

0

0

1

2

3

7

5

19

1

1

3

6

13

1

3

6

13

18

2

2

10

32

1

5

16

45

3

3

24

1

8

40

4

4

1

12

Therefore:
x^{5}+2x^{4}+3x^{3}+7x^{2}+5x+19≡k^{(5)}+12k^{(4)}+40k^{(3)}+45k^{(2)}+18k^{(1)}+19k
(0)
Using Synthetic Division, we can
comparitively easily determine the factorial form of a polynomial.
Later we shall see that using a table of Stirling Numbers
of the Second Kind, we can convert them even more easily.

Factorials
with Negative Powers

We can sometimes convert a fraction to negative factorials. For
instance, we wish to convert the following:
[2.01]
We proceed by equating coefficients, after assuming:
[2.02]
Multiplying throughout by (k+1)(k+2)(k+3):
[2.03]
Note the coefficient of k^{2} on the LHS is 0, so
A=0
Equating coefficients of k:
1=B
And equating the constant:
1=3B+C
As B=1, C=-2
We can rewrite 2.02 supplying the values for the constants, A, B, C, so:
[2.04]
Using the symbols for factorials, we can write 2.01 expressed as
factorial polynomials.
[2.05] ■

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