$k$-Fibonacci and $k$-Lucas numbers with the Hölder inequality

Abstract

Dujella, Jak\v setić and Pe\v carić recently established in \cite{DJP}, a chain of power-sum inequalities, built from Hölder's and Cauchy's inequalities and their converse forms, and applied it to the Fibonacci sequence using the identities ${\sum }_{i=1}^n{F}_i^2=F_n{F}_{n+1}$ and ${\sum }_{i=1}^n{F}_i{F}_{i+1}=F_{n+1}^2-\frac{1+(-1{)}^n}{2}$. We show that this machinery applies uniformly across the one-parameter family of $k$-Fibonacci and $k$-Lucas numbers of Falcón and Plaza, via the generalized identities \begin{align*} \sum_{i=1}^nL_{k,i}^2=\frac{L_{k,n}L_{k,n+1}-2k}{k},\qquad \sum_{i=1}^nF_{k,i}^2=\frac{F_{k,n}F_{k,n+1}}{k}, \end{align*} and \begin{align*} \sum_{i=1}^nL_{k,i}L_{k,i+1}=\frac{L_{k,n+1}^2}{k}-k+\bigl((-1)^n-1\bigr)\left(\frac{2}{k}+\frac{k}{2}\right), \end{align*} which recover the classical Fibonacci and Lucas identities at $k=1$. We further exploit the cross-identity $F_{k,i}{L}_{k,i}=F_{k,2i}$, valid for every $k\ge 1$, to obtain a Cauchy-Schwarz-type inequality linking the two families that has no counterpart in the Fibonacci-only setting. At $k=1$, this specializes to a fully explicit elementary inequality between ordinary Fibonacci and Lucas numbers, alongside the corresponding Hölder and Cauchy-conversion refinements we obtain for ordinary Lucas numbers.

arXiv comment14 pages