Bernstein Basis Functions Related to Combinatorial Sums Involving Binomial Coefficients and Special Numbers

By using generating functions, which contained alternative forms, we derive identities, some new and old, for the Bernstein basis functions. Integrating these identities, we also derive combinatorial sums involving binomial coefficients. We give relations between these combinatorial sums and generating functions for special numbers, we investigate relations related to finite sums of the powers of integers and alternating finite sums of the powers of integers associated with the special numbers. Finally, we give some comments, remarks and applications on our results. 2010 Mathematics Subject Classification. 14F10, 12D10, 26C05, 26C10, 30B40, 30C15, 42A38, 44A10.