Generalization of the Lie-Trotter product formula for q-exponential operators

The Lie-Trotter formula e ((A) over cap + (B) over cap) = lim (N --> propor tional to) (e ((A) over cap / N) e ((B) over cap / N))(N) is of great utili ty in a variety of quantum problems ranging from the theory of path integra ls and Monte Carlo methods in theoretical chemistry, to many-body and therm ostatistical calculations. We generalize it for the q-exponential function e(q)(x) = [1 + (1 - q)x]((1/(1-q))) (with e(1)(x) = e(x)), and prove e(q) ( (A) over cap + (B) over cap (1 - q)[(A) over cap (B) over cap + (A) over ca p (B) over cap]/2 = lim (N --> infinity) {[e(1-(1 - q)N) ((A) over cap / N) ] [e(1-1(1-q)N) ((B) over cap / N]}(N). This extended formula is expected t o be similarly useful in the nonextensive situations. (C) 1999 Elsevier Sci ence B.V. All rights reserved.