Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 42 min ago
PreprintarXiv licencearXiv

Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters

  • Drishya T. Das
  • Saheli Dutta
  • Shanta Laishram
  • Saranya G. Nair
Open at sourcePDFDOICite

Abstract

For a positive integer n, we study the irreducibility of the generalised Laguerre polynomials Ln(a)(x) = 1/n!∑j=0n binomnj ((n+a)!)/((j+a)!)(−1)j xj, where a ∈ ℤ≥ 0. Earlier works established that these polynomials are irreducible over ℚ for 0 ≤ a ≤ 50. In this paper, we substantially strengthen these results by establishing that Ln(a)(x) are irreducible for all a ≤ 1000. We derive sharper constraints on potential integer roots and make systematic use of the geometry of the associated Newton polygons. This approach allows us to extend the irreducibility range significantly while keeping explicit computations to a minimum.

Related

  1. The Odlyzko-Poonen Conjecture on Irreducibility of Random Polynomials
    Constantin Kogler
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryProbabilityarXiv
  2. Catalan polynomials for the Wishart-Laguerre β-ensemble
    Bei Kang, Lu-Yao Wang, Ke Wu +1
    Physics Letters BPeer-reviewedCC BY 4.0Full textStatisticsCrossref
  3. Weak Hurwitz stability of extended Turán expressions for Stirling--Whitney--Riordan and Laguerre polynomials
    Wenle Shi
    arXivPreprintarXiv licenceFull textAnalysis & PDEsCombinatorics & discrete mathematicsarXiv
  4. Catalan polynomials for the Wishart-Laguerre β-ensemble
    Bei Kang, Lu-Yao Wang, Ke Wu +1
    arXivPreprintarXiv licenceFull textMathematical physicsHigh-energy & particle physicsarXiv
  5. Irreducibility of Full Multivariate Partial-Twuality Polynomials
    Qingying Deng, Chaoyang Zhang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{das2026irreducibility,
  title = {{Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters}},
  author = {Drishya T. Das and Saheli Dutta and Shanta Laishram and Saranya G. Nair},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10614},
  eprint = {2610.10614},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10614},
}
RIS
TY  - UNPB
TI  - Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters
AU  - Drishya T. Das
AU  - Saheli Dutta
AU  - Shanta Laishram
AU  - Saranya G. Nair
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10614
UR  - https://arxiv.org/abs/2610.10614
AB  - For a positive integer n, we study the irreducibility of the generalised Laguerre polynomials L_n^(a)(x) = 1/n!∑_j=0^n binomnj ((n+a)!)/((j+a)!)(−1)^j x^j, where a ∈ ℤ_≥ 0. Earlier works established that these polynomials are irreducible over ℚ for 0 ≤ a ≤ 50. In this paper, we substantially strengthen these results by establishing that L_n^(a)(x) are irreducible for all a ≤ 1000. We derive sharper constraints on potential integer roots and make systematic use of the geometry of the associated Newton polygons. This approach allows us to extend the irreducibility range significantly while keeping explicit computations to a minimum.
ER  -