Skip to content
DraconiticResearch
Search Sources API
197,761 studies31 sourcesUpdated just now
PreprintCC BY 4.0arXiv

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

  • Herbert Batte
  • Prosper Kaggwa
Open at sourcePDFDOICite

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 ∑i=1nFi2=FnFn+1 and ∑i=1nFiFi+1=Fn+12−(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 ∑i=1nLk,i2=(Lk,nLk,n+1−2k)/k, ∑i=1nFk,i2=(Fk,nFk,n+1)/k, and ∑i=1nLk,iLk,i+1=(Lk,n+12)/k−k+((−1)n−1)(2/k+k/2), which recover the classical Fibonacci and Lucas identities at k=1. We further exploit the cross-identity Fk,iLk,i=Fk,2i, valid for every k≥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.

Related

  1. Infinite Series About Reciprocal Products of Fibonacci and Lucas Numbers
    Marta Na Chen, Wenchang Chu
    AxiomsPeer-reviewedCC BY 4.0Algebra & number theoryCrossref
  2. Two Primes and Five Mixed Fibonacci-Lucas Summands
    Zhao Shen, Chenghe Xu
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Monogenity of the irreducible factors of (k,t)-Fibonacci and (k,t)-Lucas polynomials
    Michail Karatarakis, Sumandeep Kaur
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. The local Dold congruence for Fibonacci and Lucas sequences
    Sompong Chuysurichay, Sawian Jaidee, Chatchawan Panraksa +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Cancellation profiles of Fibonacci-Lucas zeta sums
    Payam Danesh
    arXivPreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{batte2026kfibonacci,
  title = {{$k$-Fibonacci and $k$-Lucas numbers with the Hölder inequality}},
  author = {Herbert Batte and Prosper Kaggwa},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33573},
  eprint = {2609.33573},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33573},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - k-Fibonacci and k-Lucas numbers with the Hölder inequality
AU  - Herbert Batte
AU  - Prosper Kaggwa
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33573
UR  - https://arxiv.org/abs/2609.33573
AB  - 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 ∑_i=1^nF_i^2=F_nF_n+1 and ∑_i=1^nF_iF_i+1=F_n+1^2−(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 ∑_i=1^nL_k,i^2=(L_k,nL_k,n+1−2k)/k, ∑_i=1^nF_k,i^2=(F_k,nF_k,n+1)/k, and ∑_i=1^nL_k,iL_k,i+1=(L_k,n+1^2)/k−k+((−1)^n−1)(2/k+k/2), which recover the classical Fibonacci and Lucas identities at k=1. We further exploit the cross-identity F_k,iL_k,i=F_k,2i, valid for every k≥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.
ER  -