Butterworth passive filters normalized values equation
Start here with the Cauer topology:

https://en.wikipedia.org/wiki/Butterworth_filter

Notice that these are the same as your equations. Also notice that the equations don't specify the frequency either, it is assumed to be 1 rad/sec. The diagram doesn't show the source and load resistor, but the text says that they are unity (1).

Now go here to the Impedance Scaling section:

https://en.wikipedia.org/wiki/Prototype_filter

This shows you how to scale to your resistance and cutoff frequency.