On an Identity by Bruckman and Good

TL;DR

This paper presents a variation of a mathematical identity by Bruckman and Good, enabling derivation of sums involving Fibonacci and Lucas numbers, including those with arithmetic progression indices, and generalizes Millin series.

Contribution

It introduces a new variation of the Bruckman-Good identity and applies it to derive and generalize sums involving Fibonacci, Lucas numbers, and Millin series.

Findings

Derived sums involving Fibonacci and Lucas numbers with arithmetic progression indices

Generalized Millin series using the new identity

Provided new formulas expanding known series and identities

Abstract

This paper introduces a variation on an identity by Bruckman and Good. Using this identity, we are able to derive various well-known sums involving reciprocals of Fibonacci and Lucas numbers, including the case when the indices form an arithmetic progression. Moreover, we provide generalizations of the Millin series.