Skip to content
DraconiticResearch
Search Sources API
201,818 studies31 sourcesUpdated 16 min ago
PreprintCC BY 4.0arXiv

Holonomic sequences arising from meromorphic modular forms

  • Kam Cheong Au
Open at sourcePDFDOICite

Abstract

We introduce a systematic framework for constructing holonomic recurrences from meromorphic modular forms. Its central ingredient is a modular analogue of creative telescoping, which turns finite-dimensional quotients of spaces of meromorphic modular forms into explicit recurrence operators. This provides a common explanation for several previously isolated examples: it recovers Apéry's recurrence, Beukers' modular constructions and Zagier's seven sporadic recurrences. The framework also produces a large collection of new examples across different weights and groups. The modular origin of these sequences also confers them with non-trivial arithmetic properties, such as denominator bounds, irrationality statements, and p-adic congruences.

Related

  1. On the abundance of rigid meromorphic cocycles for quadratic forms in four variables
    Lennart Gehrmann, Sören Sprehe
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. Meromorphic modular differential equations for the elliptic genus of Calabi--Yau varieties
    Dmitrii Adler, Valery Gritsenko
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. On the structure of polynomial multicorrelation sequences arising from nilpotent actions
    Jovan Andreevski, Felipe Hernández
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsDynamical systemsarXiv
  4. On the Diophantine Equations Arising from Three-Term b-Concatenations in Pell-Type Sequences
    Kouèssi Norbert Adédji, Marija Bliznac Trebješanin, Jelena Pleština
    arXivPreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryarXiv
  5. Partition statistics and universal mixed mock modular forms
    Koustav Banerjee, Kathrin Bringmann
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{au2026holonomic,
  title = {{Holonomic sequences arising from meromorphic modular forms}},
  author = {Kam Cheong Au},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17240},
  eprint = {2610.17240},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17240},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Holonomic sequences arising from meromorphic modular forms
AU  - Kam Cheong Au
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17240
UR  - https://arxiv.org/abs/2610.17240
AB  - We introduce a systematic framework for constructing holonomic recurrences from meromorphic modular forms. Its central ingredient is a modular analogue of creative telescoping, which turns finite-dimensional quotients of spaces of meromorphic modular forms into explicit recurrence operators. This provides a common explanation for several previously isolated examples: it recovers Apéry's recurrence, Beukers' modular constructions and Zagier's seven sporadic recurrences. The framework also produces a large collection of new examples across different weights and groups. The modular origin of these sequences also confers them with non-trivial arithmetic properties, such as denominator bounds, irrationality statements, and p-adic congruences.
ER  -