New Polynomial Identities and Some Consequences

TL;DR

The paper introduces new polynomial identities using elementary methods, generalizes known identities, and develops a scheme for deriving related combinatorial identities, with implications for combinatorics.

Contribution

It presents novel polynomial identities derived via elementary techniques and generalizes classical combinatorial identities, offering a new framework for related identity derivations.

Findings

Derived two new polynomial identities using elementary methods.

Generalized Frisch's and Klamkin's identities.

Established a scheme for deriving combinatorial identities.

Abstract

Using an elementary approach involving the Euler Beta function and the binomial theorem, we derive two polynomial identities; one of which is a generalization of a known polynomial identity. Two well-known combinatorial identities, namely Frisch's identity and Klamkin's identity, appear as immediate consequences of the polynomial identities. We subsequently establish several combinatorial identities, including a generalization of each of Frisch's identity and Klamkin's identity. Finally, we develop a scheme for deriving combinatorial identities associated with polynomial identities of a certain type.