Bandpass general equation to difference equation
The transfer function of a continuous-time second-order band-pass filter is given by

$$H(s)=\frac{\frac{\omega_0}{Q}s}{s^2+\frac{\omega_0}{Q}s+\omega_0^2}\tag{1}$$

where $\omega_0$ is the center frequency in radians per second, and $Q$ is the quality factor. For $Q\gg 1$, the term $\omega_0/Q$ closely approximates the $3$ dB bandwidth $W$ (in radians per second). Note that the condition $Q\gg 1$ is equivalent to $W\ll\omega_0$.

If you want to transform the transfer function $(1)$ to the discrete-time domain, there are several possibilities. The two most commonly used ones are the bilinear transform and the impulse invariance method. Both of them are covered extensively in the literature and on the web (including this site). Note that the bilinear transform introduces frequency warping, so you have to pre-warp the parameters of the continuous-time filter in such a way that after the transformation the discrete-time filter has the desired properties. A very good explanation of IIR filter design using the bilinear transform can be found in chapter 11 of the (freely available) book Introduction to Signal Processing by S.J. Orfanidis.

You can find specific formulas for second-order discrete-time filters derived from the bilinear transform in the Audio-EQ-Cookbook. Also take a look at this question and its answers.