Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Nick Vorobtsov

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.

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

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.
Paper Structure (4 sections, 6 theorems, 16 equations)

This paper contains 4 sections, 6 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Basic Definitions and Lemmas
  3. Main Results
  4. Conclusion

Key Result

Lemma 1

Let $\alpha = \frac{1 + \sqrt{5}}{2}$ and $\beta = \frac{1 - \sqrt{5}}{2}$ be the roots of the characteristic equation $x^2 - x - 1 = 0$. Then the explicit formulas are valid: Since $\alpha \beta = -1$, the identity $\alpha^n \beta^n = (-1)^n$ holds.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1: Binet's Formula koshy2001
  • Lemma 2: Waring's Formulas lidl1997
  • Lemma 3: d'Ocagne's Identity koshy2001
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 2 more