Advanced z-transform

In mathematics and signal processing, the advanced z-transform is an extension of the z-transform, to incorporate ideal delays that are not multiples of the sampling time. It takes the form
where
It is also known as the modified z-transform.
The advanced z-transform is widely applied, for example to accurately model processing delays in digital control.
Properties.
If the delay parameter, "m", is considered fixed then all the properties of the z-transform hold for the advanced z-transform.
Example.
Consider the following example where formula_8:
If formula_10 then formula_11 reduces to the transform
which is clearly just the "z"-transform of formula_13.