A Family of the Zeckendorf Theorem Related Identities

TL;DR

This paper introduces a family of identities related to Zeckendorf representations for recursive sequences, providing new insights into Fibonacci and silver ratio representations through bijections and algebraic equalities.

Contribution

It presents novel identities for recursive sequences connected to Zeckendorf representations and establishes bijections and algebraic equalities involving Fibonacci and silver ratios.

Findings

Identities for recursive sequences related to Zeckendorf representations

Bijections between number representations and ratios

Representation of Fibonacci and silver ratio sums

Abstract

In this paper we present a family of identities for recursive sequences arising from a second order recurrence relation, that gives instances of Zeckendorf representation. We prove these results using a special case of an universal property of the recursive sequences. In particular cases we also establish a direct bijection. Besides, we prove further equalities that provide a representation of the sum of $(r+1)$-st and $(r-1)$-st Fibonacci number as the sum of powers of the golden ratio. Similarly, we show a class of natural numbers represented as the sum of powers of the silver ratio.