Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 47 min ago
PreprintCC BY 4.0arXiv

Polynomial Identities for Generalized Fibonacci Sequences

  • Joey Glampe
  • Aryaman Joshi
  • Swagat Malla
  • Matthew Richey
  • Matthew Wright
Open at sourcePDFDOICite

Abstract

The Fibonacci, Lucas, and Pell sequences satisfy certain families of ``polynomial'' identities. While a few members of these families are well-known, we are not able to find a general statement of these identities in the mathematical literature. We show that such polynomial identities exist for many generalized Fibonacci sequences, and we classify sequences that have these identities obtained by two distinct approaches. One approach uses the Girard-Waring formula, while the other relies on Binet's formula and linear algebra.

Related

  1. Hurwitz Stability of Generalized Turán Expressions of Polynomial Sequences
    Huihua Gao, Xin-Bei Liu
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. The local Dold congruence for Fibonacci and Lucas sequences
    Sompong Chuysurichay, Sawian Jaidee, Chatchawan Panraksa +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. On random Binomial-Fibonacci sequences
    Bernard Bercu, Fabrice Gamboa
    arXivPreprintarXiv licenceFull textDynamical systemsProbabilityarXiv
  4. Hopf algebra polynomial identities in the Sweedler algebra
    Alexey Gordienko, Maria Morozova
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Asymptotics for partitions over the Fibonacci numbers and related sequences, II
    Michael Coons, Simon Kristensen, Mathias L. Laursen
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{glampe2026polynomial,
  title = {{Polynomial Identities for Generalized Fibonacci Sequences}},
  author = {Joey Glampe and Aryaman Joshi and Swagat Malla and Matthew Richey and Matthew Wright},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15131},
  eprint = {2610.15131},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15131},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Polynomial Identities for Generalized Fibonacci Sequences
AU  - Joey Glampe
AU  - Aryaman Joshi
AU  - Swagat Malla
AU  - Matthew Richey
AU  - Matthew Wright
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15131
UR  - https://arxiv.org/abs/2610.15131
AB  - The Fibonacci, Lucas, and Pell sequences satisfy certain families of ``polynomial'' identities. While a few members of these families are well-known, we are not able to find a general statement of these identities in the mathematical literature. We show that such polynomial identities exist for many generalized Fibonacci sequences, and we classify sequences that have these identities obtained by two distinct approaches. One approach uses the Girard-Waring formula, while the other relies on Binet's formula and linear algebra.
ER  -