Proof of Decreasing Exponential Factorials properties

**greenday1209** · May 1st 2011, 06:44 AM

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

**CaptainBlack** · May 1st 2011, 10:10 PM

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

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