In this paper, we first provide some functional equations of the generating 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 involving 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 combinatorial identities, the Catalan numbers, and the harmonic numbers. Moreover, by applying the classical Young inequality, we derive a combinatorial inequality related to beta polynomials and combinatorial sums. We also give another inequality for the Catalan numbers.
Combinatorial sums, Binomial identities, Generating functions, Functional equations, Beta polynomials, Beta function, Gamma function, Bernstein basis functions, Catalan numbers, Harmonic numbers, Young inequality