New proofs of some theorems for binomial transform and Fibonacci powers

TL;DR

This paper provides new proofs for theorems related to binomial transforms and Fibonacci powers, specifically addressing sums involving Fibonacci numbers raised to powers congruent to 1 or 3 mod 4, building on previous work by Wessner and Hoggatt.

Contribution

It introduces alternative proofs for existing theorems on Fibonacci power sums using methods from Boyadzhiev's work, and extends results to cases with powers congruent to 1 or 3 mod 4.

Findings

New proofs for Fibonacci power sum theorems.

Extended results to cases p ≡ 1 or 3 mod 4.

Alternative presentation of Wessner's results.

Abstract

Our aim in writing this paper is to answer to both V. E. Hoggatt, JR \cite{hogg} and Wessner\cite{wess} on the next question: find $\sum_{k=0}^n\binom{n}{k}F_{[k]}^p$, for the case $p\equiv 1\, mod\, 4$ and $p\equiv 3\, mod\, 4$. \par The case $p\equiv 0\, mod \,4$ and $p\equiv 2\, mod\, 4$, Wessner has given an answer. In particular, we give another presentation, another proof of the paper of Wessner. Our method use, essentially, the paper of Boyadzhiev\cite{boy}