A look over the Blue Note article on Wikipedia that Shevliaskovic linked talks a bit about the tuning theory behind Blue Notes, so I'd like to expand on that, as you mentioned wanting a "Mathematical" definition of these notes.
The article states that in order to overcome tuning hardships in keyboard creation in the 18th century, Equal Temperament was instituted and is the standard today. Rather than following the Harmonic Series (harmony in nature) or Pythagorean Temperament (the circle of fifths), it divided the octave into equal parts. That means that each note we hear in Equal Temperament is slightly out of tuning for "natural" harmony, which is what the African immigrants would have been accustomed to.
This brings us to a bit of Just Intonation, which is the practice of tuning notes to whole number ratios. Intervals between notes in equal temperament are expressed with exponents. For example, a minor third in ET would be 23/12 and a major third would be 24/12. There are many intervals in Just Intonation that are considered major thirds and minor thirds, but we'll look at two of the most basic ones. The major third is 5/4 and the minor third is 6/5. Without too much discussion into specifics, let's consider these the "natural" harmony that is ideal to our ears. You can see how they are tuned differently (in cents):
Major 3rd 386.31¢ 400¢
Minor 3rd 315.64¢ 300¢
You'll see that the M3 is a little flat from Equal Temperament and the m3 is a little sharp. This is one way that a "blue" third may have pulled inward toward the half-way note that it is. It is also worth mentioning "neutral" intervals, which fall right in between major and minor intervals. 11/9, 60/49, 49/40, 27/22, and 16/13 are all examples of neutral third and they clock in at 347.4¢, 350.61¢, 351.33¢, 354.54¢, and 359.47¢, respectively.
This isn't all to say that these intervals are exactly what a blue note is, just how it may be derived. I can't seem to find an scholarly articles examining blues singers or players, so I guess the specifics of this mystery will still go unsolved.