  In this paper, we found new q-binomial formula for Q-commutative operators.
Expansion coefficients in this formula are given by q-binomial coefficients
with two bases (q,Q), determined by Q-commutative q-Pascal triangle. Our
formula generalizes all well-known binomial formulas in the form of Newton,
Gauss, symmetrical, non-commutative and Binet-Fibonacci binomials. By our
non-commutative q-binomial, we introduce q-analogue of function of two
non-commutative variables, which could be used in study of non-commutative
q-analytic functions and non- commutative q-traveling waves.
