New Binomial Identities for Fibonacci, Lucas, and Generalized Fibonacci Sequences with Multiple Indices

TL;DR

This paper introduces novel binomial identities that relate Fibonacci, Lucas, and generalized Fibonacci sequences with multiple indices to powers of Lucas numbers and binomial coefficients, using symmetric polynomial techniques.

Contribution

It develops new identities for these sequences involving multiple indices, expanding the mathematical understanding of their structure and relationships.

Findings

Derived identities express sequence terms via binomial coefficients and Lucas numbers.

Applied symmetric polynomials to classical Binet's formula for sequence analysis.

Provided binomial expansions for generalized Fibonacci sequences combining Pascal's triangle elements.

Abstract

This paper presents new identities expressing the terms of Fibonacci, Lucas, and generalized Fibonacci sequences with multiple indices through powers of Lucas numbers and binomial coefficients. The obtained formulas rely on the application of symmetric polynomials (Waring's formulas) to the classical Binet's formula. Particular attention is given to the binomial expansion for the generalized Fibonacci sequence, which structurally combines two adjacent binomial coefficients from Pascal's triangle.