A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance:

$y[n] = x[n] + y[n-1]$

$Y(z) = X(z) + Y(z) \cdot z^{-1}$

$H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1-z^{-1}}$

and now the complex plot of $H(z)$ gives us plenty of useful information - in this case, we have a pole at 1, corresponding to DC.

Likewise, we can look at "scale-difference" equations, where a fractional argument to the arithmetic function returns 0 by convention. Now it's the Dirichlet transform $F(z) = \sum_n (f[n] \cdot n^{-z} )$ that's helpful, keeping the variable as z for notational clarity:

$y[n] = x[n] + y[\frac{n}{2}]$

$Y(z) = X(z) + Y(z) \cdot 2^{-z}$

$H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1-2^{-z}}$

and now the complex plot of $H(z)$ likewise gives us plenty of useful information - in this case, we have poles at $-\frac{2 \pi i n}{\log(2)}, n \in \mathbb{Z}$.

My question - **Is there some sort of transform useful for both "shift-" and "scale-"difference equations?**

Here's an example of the sort of equation I'm talking about:

- $y[n] = x[n] + y[\frac{n}{2}] + y[\frac{n-1}{2}]$

It would be great if the above could somehow, for some sort of transform, turn into something along the lines of

$Y(z) = X(z) + Y(z) \cdot 2^{-z} + Y(z) \cdot z^{-\frac{1}{2}} \cdot 2^{-z}$

$H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1 - 2^{-z} - z^{-\frac{1}{2}} \cdot 2^{-z}}$

Or whatever. Some quick thinking will show that the above obviously doesn't work ("scale" and "shift" operations don't commute like that), but this illustrates what I'm getting at.

**Does something like this exist?**