New proofs of some theorems for binomial transform and Fibonacci powers

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}