# An Identity for Second Order Sequences Obeying the Same Recurrence Relation

## TL;DR

This paper derives a general identity linking second-order linear recurrence sequences with the same recurrence but different initial conditions, unifying and extending many known identities across various classical sequences.

## Contribution

It introduces a new, general identity for second-order recurrence sequences that encompasses and extends existing identities, with applications to well-known sequences.

## Key findings

- Unified identity for second-order sequences
- Derivation of binomial and summation identities
- Examples with Fibonacci, Pell, Jacobsthal sequences

## Abstract

We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are developed. Illustrative examples are drawn from Fibonacci, Fibonacci-Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences and their generalizations. Our new results subsume previously known identities.