Abstract. We study a new sequence of Bernstein-type operators on the $d$-dimensional simplex. The binomial coefficients considered in the classical definition are substituted with more general ones satisfying a similar recursive formula and this produces a first-order perturbation term in the Voronovskaja formula. As a consequence of the Trotter's theorem, we may approximate the solutions of suitable second-order degenerate parabolic problems, which are of particular interest as gene frequency models in population genetics. With respect to the classical Bernstein operators, these new sequences of operators allow to consider new factors as mutation, migration and selection in the associated diffusion processes.

AMS Subject Classification (1991): 41A36, 60J70, 34A45, 92D15

Keyword(s): Bernstein-type Operators, Positive Approximation, Population Genetics

Received November 15, 2000. (Registered under 2836/2009.)