On the Brousseau sums $\sum_{i=1}^n i^p F_i$

TL;DR

This paper introduces new convolution formulas involving binomial coefficients to derive explicit sums of powers multiplied by Fibonacci numbers, connecting to prior research and providing direct computation methods.

Contribution

It presents novel convolution formulas for Fibonacci sums with powers, offering direct computation methods and linking to existing literature.

Findings

Derived new convolution formulas involving binomial coefficients.

Provided explicit formulas for sums of powers times Fibonacci numbers.

Connected new formulas to previous research in the field.

Abstract

We start with new convolution formulas for $F_n - n^p$ involving only the binomial coefficients. Then, we use those to find direct formulas for the sums $\sum_{i=1}^n i^p F_{n-i}$ and $\sum_{i=1}^n i^p F_i$, and we show how our formulas connect to work in earlier papers by Ledin, Brousseau, Zeitlin, Adegoke, Shannon and Ollerton, and Kinlaw, Morris, and Thiagarajan.