For a discrete-time LTI system, the transfer function is the ratio of z-transformed output to input under zero initial conditions:

$$H(z)=\frac{Y(z)}{X(z)}.$$

A linear difference equation becomes algebraic after the z-transform because delays become powers of $z^{-1}$.

For example,

$$y[n]-ay[n-1]=bx[n]$$

gives

$$Y(z)-az^{-1}Y(z)=bX(z),$$

so

$$H(z)=\frac{b}{1-az^{-1}}.$$

The transfer function is also the z-transform of the impulse response:

$$H(z)=\mathcal Z{h[n]}.$$

When the region of convergence includes the unit circle, evaluating

$$z=e^{i\omega}$$

gives the discrete-time frequency response $H(e^{i\omega})$.

As in continuous time, the transfer function describes zero-state input-output behavior rather than a particular stored initial state.