Binomial sum relations involving Fibonacci and Lucas numbers

TL;DR

This paper explores new relations between binomial sums involving Fibonacci and Lucas numbers, revealing connections with various binomial coefficients and rediscovering some known relations.

Contribution

It introduces novel relations between binomial sums and Fibonacci/Lucas numbers, including sums with multiple binomial coefficients, expanding understanding of these mathematical structures.

Findings

New relations between binomial sums and Fibonacci/Lucas numbers

Connections between sums with two and three binomial coefficients

Rediscovery of some previously known relations

Abstract

In this paper, we introduce relations between binomial sums involving (generalized) Fibonacci and Lucas numbers, and different kinds of binomial coefficients. We also present some relations between sums with two and three binomial coefficients. In the course of exploration we rediscover a few relations presented as problem proposals.