How to sum powers of balancing numbers efficiently

TL;DR

This paper presents an efficient method to compute sums of powers of balancing numbers by expressing them as linear combinations of scaled balancing numbers and employing generating functions.

Contribution

It introduces a novel approach to sum powers of balancing numbers explicitly using their Binet formula and generating functions.

Findings

Explicit formulas for partial sums of powers of balancing numbers.

Efficient computation method for sums involving balancing numbers.

Application of generating functions to simplify summations.

Abstract

Balancing numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of balancing numbers can be summed explicitly. For this, as a first step, a power $B_n^l$ is expressed as a linear combination of $B_{mn}$. The summation of such expressions is then easy using generating functions.