Triangle formulas in the complex plane

where R(f; z1) is the remainder at zi due to linear interpolation to f(z) at Z2 and Z3. This formula was subsequently found in Approximation: Theory and Practice by I. J. Schoenberg (Notes on a series of lectures at Stanford University, 1955) and correspondence with Schoenberg revealed that it was obtained by him and T. Motzkin in 1951-1952 during one of their numerous luncheon conversations. In the set of notes just referred to, the formula was derived by applying the HermiteGenocchi integral representation for divided differences, and this approach suggests certain generalizations to nth order divided differences. The object of the present paper is to publicize this interesting formula, to give an alternate proof of it, and to derive a number of consequences and results related to the alternate approach.