Thread: Proof of Decreasing Exponential Factorials properties

1. Proof of Decreasing Exponential Factorials properties

I need help checking the following properties (whether they're true or false):

$x^{\underline{m.n}} = \prod_{k = 0}^{m-1} (x - kn)^\underline{n}$

$x^{\underline{m.n}} = \prod_{k = 0}^{n-1} (x - km)^\underline{n}$

$x^{\underline{m.n}} = \prod_{k = 0}^{n-1} (x - km)^\underline{m}$

where $x^{\underline{m.n}}$ is a decreasing factorial exponential, i.e.:

$x^{\underline{m.n}} = \prod_{k = 0}^{m.n - 1} (x-k)$

Thanks in advance

2. Originally Posted by greenday1209
I need help checking the following properties (whether they're true or false):

$x^{\underline{m.n}} = \prod_{k = 0}^{m-1} (x - kn)^\underline{n}$

$x^{\underline{m.n}} = \prod_{k = 0}^{n-1} (x - km)^\underline{n}$

$x^{\underline{m.n}} = \prod_{k = 0}^{n-1} (x - km)^\underline{m}$

where $x^{\underline{m.n}}$ is a decreasing factorial exponential, i.e.:

$x^{\underline{m.n}} = \prod_{k = 0}^{m.n - 1} (x-k)$

Thanks in advance
Numerical experiment shows that the second is false.

If either the first or third is true then the other must also be true. Numerical experiment does not refute them.

CB