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

This preprint presents new identities for Fibonacci, Lucas, and generalized Fibonacci sequences with multiple indices, expressed through powers of Lucas numbers and binomial coefficients. The proofs use symmetric polynomials (Waring’s formulas) together with Binet’s formula. The case of generalized Fibonacci numbers is highlighted: the binomial expansion combines two adjacent binomial coefficients from Pascal’s triangle.

Also available on arXiv: https://arxiv.org/abs/2603.12150
Author’s page (interactive materials and related work): https://nvvorobtsov.github.io/