Efficient evaluation of Bernstein-Bézier coefficients of B-spline basis functions over one knot span

New differential-recurrence relations for B-spline basis functions are given. Using these relations, a recursive method for finding the Bernstein-B\'{e}zier coefficients of B-spline basis functions over a single knot span is proposed. The algorithm works for any knot sequence which guarantees that all B-spline functions are at least $C^0$-continuous. It has good numerical behavior and has an asymptotically optimal computational complexity.