Generic response of an IIR filter from its transfer function
$$Y(z)=X(z)H(z)=\frac{X(z)}{1+\sum\limits_{m=1}^N{a_m z^{-m}}}$$

means

$$\left(1+\sum\limits_{m=1}^N{a_m z^{-m}} \right)Y(z) = X(z)$$

$$Y(z) + \sum\limits_{m=1}^N{a_m Y(z) z^{-m}} = X(z)$$

$$y[n] + \sum\limits_{m=1}^N{a_m y[n-m]} = x[m]$$

$$y[n] = - \sum\limits_{m=1}^N{a_m y[n-m]} + x[m]$$