Of course 17 mm is 17 mm. Lenses cannot think by themselves so they have no idea about what the recording surface behind it might be. For all the lens knows, it could be 8x10" or 24x36 mm or DX. The lens is plain stupid and doesn't care anyway. So it cannot change its inherent focal length at will. Neither can it change its DOF properties. It is just too stupid to achieve such a feat.
Another fact is that any lens has a "crop" factor. Always. It just so happens that for some recording format, based upon tradition, the factor is unity, elsewhere it is not. Thus, we are entitled to call an 85 mm lens as being a "crop" factor 2 lens on the traditional 24x36 mm (it is the normal lens for 6x6 cm medium format), or having "crop" factor 3 for DX format, or "crop" factor 0.75 for the 4x5". All of these factors are of course nonsense and utterly useless information.
Since we *should* focus on angle of view instead of the silly "crop" factor, learn that any lens with a focal length (approximately) equal to the diagonal of the format gives 45 degrees field of view. This is what we call the "normal" for the given format. If you double the focal length, the angle of view is cut into half, if you make the focal length half of the "normal", the angle of view is doubled. This is simple math and based upon this, you can easily calculate the angle of view for any lens on any format. Or if your math capacity is impaired, just look into the viewfinder. The answer is the same.
The rationale behind "DX" or "digital" lenses has nothing to do with the size of their image circle as such, although Nikon DX lenses are designed specifically for the smaller "DX" sized sensor. They are so-called "telecentric" designs, meaning they send virtually parallel rays onto the imager. The digital chip thinks this is a very nice way to be treated, since the angle of incidence will be virtually normal (approx. 90 degrees), and accordingly it responds with better image quality, almost no geometric vignetting into the corners of the frame, and much less CA as well - the optical equivalent to a heartfelt "thank you".
A pecularity with DX lenses is that their image circle is biggest at infinity focus and is reduced when they focus closer (exactly opposite of non-telecentric lenses). So even if a DX lens can fill the entire frame of a 24x36 camera at infinity, it may not do so at close range. Many of the zoom DX lenses from Nikon can be used on "full-frame" cameras (for which the DX lenses will have "crop" factor less than unity, viz. 0.67) for some part of their focal range, but not all, for example the 18-70 DX has its largest projected image circle at its 50 mm setting but still vignettes on a 24x36 mm camera.