On Polynomial Identities for Recursive Sequences

TL;DR

This paper extends polynomial identities to Pell and Pell-Lucas sequences, generalizing Melham sums and providing new expressions for various recursive sequences using binomial theorem and known identities.

Contribution

It introduces new polynomial identities for Pell-related sequences and extends existing expressions to other recursive sequences, combining combinatorial and algebraic methods.

Findings

Extended Melham sums to Pell and Pell-Lucas sequences

Derived new polynomial identities for recursive sequences

Connected identities with binomial theorem and known Pell identities

Abstract

In this paper we extend the notion of Melham sum to the Pell and Pell-Lucas sequences. While the proofs of general statements rely on the binomial theorem, we prove some spacial cases by the known Pell identities. We also give extensions of obtained expressions to the other recursive sequences.