Hopefully the reader is familiar with the counting properties of the factorial that follows the recurrence relation of

- n!=n (n-1)!
- 1!=1.

From the above it is not hard to show that:

- 2!=2(1!)=2(1)=2
- 3!=3(2!)=3(2)=6
- 4!=4(3!)=24
- 5!=5(4!)=120
- 6!=6(5!)=720

Perhaps a bit complexing is the value of the factorial at 0, this is best viewed as

- 1!=1(0!)
- 1=1(0!).

It follows naturally that 0!=1. {author note, ref linked.com group posting via India.}

As gleanned from the unit shifted gamma funciton, .i.e., the factorial is never zero, but has poles at the negative integers {-1,-2,-3, … }.

It can be shown that the reciporical of the factorial function is entire. In addition, it has zeros at the negative integers. It is well known that

The reciporical factioral can be shown to satisfy the recurrence of

.

*ad infinitium*. Therefore,

,

where the product is taken over the naturals.