Combinatorial identities of three complex parameters and their basic applications

Inspired by a problem proposal recently published in the journal The Fibonacci Quarterly we offer a generalization consisting of two combinatorial identities involving three complex parameters. These identities turn out to be immensely rich. We demonstrate this by providing basic applications to four different fields: polynomial identities, trigonometric identities, identities involving Horadam numbers, and combinatorial identities. Many of our findings will generalize existing results.