Hi OSTexo, well I'm not really sure if it matches what you're asking for, but I guess the technical answer is yes  every NURBS curve does have a mathematical equation which represents the curve just because that's the basic mechanism for how NURBS curves work.
But this is true of any NURBS curve, even a line segment, it's not something unique to G1, G2, or G3 curves.
Whether things are G1, G2, or G3 where they touch means whether they have certain geometric properties equal at the touching point, for G1 it means tangents equal at the touching point, G2 means curvature equal at the touching point, and G3 means rate of change of curvature equal at the touching point.
The "Gness" of things has to do with how smoothly things touch each other, it doesn't really have to do with the specific overall shape itself, so I'm not quite so sure that the question you're asking is really properly formed... Talking about G1, G2 kind of implicitly means you're talking about some kind of meeting point between different pieces of things.
G1, G2, or G3 does not itself imply any specific shape other than an equality of geometric properties at just one individual meeting point.
Not sure if that helps you or not!
Also it's not unusual for NURBS curves to be produced as the result of a fitting operation, where various things like intersections are guided by some higher level operation like intersection between surfaces but represented with a general bendy curve that just has enough points in it so that the bendy curve is within some given tolerance of the ideal "procedural" based curve. Procedural means something based off of some higher level process. If all you have is the fitted end result, it's not really very easy to reverse engineer the original underlying process behind it, except maybe if you have some special case knowledge of what was going on... Not sure if this applies to what you're asking about.
 Michael
