Generalized Popoviciu Expansions for Bernstein Polynomials of a Rational Module

We study Bernstein polynomials for simple nonsmooth rational module functions. show that these can be represented as special sums of regular structure. For historical reasons, the representations found are naturally called “generalized Popoviciu expansions.” To write generalized expansions, we develop a formalism based on combinatorial calculations. Based formulas obtained, propose complete description character convergence studied in complex plane. also discuss relationship expansions with distribution zeros polynomials. In final part paper, present additional new relations function.