Transfer function of a nonhomogeneous difference equation

Question

Consider the following difference equation:

$y_k=\alpha y_{k-1}+\beta x_k$

The transfer function for this is given by:

$\displaystyle\frac{Y(z)}{X(z)}=\frac{\beta}{1-\alpha z^{-1}} = \frac{\beta}{z-\alpha}$

How does one calculate the transfer function for

$y_k=\alpha y_{k-1}+\beta x_k + \gamma$?

Answer 1

The last difference equation is not a linear system due to the addition of the constant $\gamma$, therefore it does not have a transfer function.