Combinatorial sums and binomial identities associated with the Beta-type polynomials

Abstract

In this paper, we first provide some functional equations of the gener ating functions for beta-type polynomials. Using these equations, we derive various identities of the beta-type polynomials and the Bernstein basis functions. We then obtain some novel combinatorial identities in volving binomial coefficients and combinatorial sums. We also derive some generalizations of the combinatorics identities which are related to the Gould's identities and sum of binomial coefficients. Next, we present some remarks, comments, and formulas including the combi natorial identities, the Catalan numbers, and the harmonic numbers. Moreover, by applying the classical Young inequality, we derive a combi natorial inequality related to beta polynomials and combinatorial sums. We also give another inequality for the Catalan numbers.