Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 59 min ago
PreprintarXiv licencearXiv

Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function

  • Andrius Grigutis
  • Yuri Matiyasevich
  • Lukas Turčinskas
Open at sourcePDFDOICite

Abstract

We prove four explicit continued fraction representations for the Lerch transcendent Φ(z, s, M+1), where ℜ M>0 and (z,s)∈{(−1,1),(−1,2),(1,2),(1,3)}. All of the continued fractions have unit partial numerators, while partial denominators depend on the parameter M. The proofs combine equivalent transformations of continued fractions, generalized hypergeometric functions, three-term recurrences, and asymptotic analysis of minimal solutions. For z=1, the corresponding representations give continued fractions for the values of the Hurwitz zeta function ζ(2, M+1) and ζ(3, M+1).

Related

  1. An elementary treatment of Ramanujan's elementary continued fractions
    Gaurav Bhatnagar
    arXivPreprintCC BY-SA 4.0Full textAlgebra & number theoryAnalysis & PDEsarXiv
  2. Continued Fractions of Polynomial Type: Theory and Encyclopedic Dictionary
    Henri Cohen
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. Optimal local convergence criteria for integer and Gaussian integer continued fractions
    Ian Short, Margaret Stanier, Matty van Son +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryAnalysis & PDEsarXiv
  4. The (p,q)-Analogues of Bi-Stirling Eulerian Polynomials via continued fractions
    Chao Xu, Jiang Zeng
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Two-point Padé approximation and patterns in continued fractions for (Fn+1/Fn)k
    Evan M. O'Dorney
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{grigutis2026four,
  title = {{Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function}},
  author = {Andrius Grigutis and Yuri Matiyasevich and Lukas Turčinskas},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08595},
  eprint = {2609.08595},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08595},
}
RIS
TY  - UNPB
TI  - Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function
AU  - Andrius Grigutis
AU  - Yuri Matiyasevich
AU  - Lukas Turčinskas
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08595
UR  - https://arxiv.org/abs/2609.08595
AB  - We prove four explicit continued fraction representations for the Lerch transcendent Φ(z, s, M+1), where ℜ M>0 and (z,s)∈{(−1,1),(−1,2),(1,2),(1,3)}. All of the continued fractions have unit partial numerators, while partial denominators depend on the parameter M. The proofs combine equivalent transformations of continued fractions, generalized hypergeometric functions, three-term recurrences, and asymptotic analysis of minimal solutions. For z=1, the corresponding representations give continued fractions for the values of the Hurwitz zeta function ζ(2, M+1) and ζ(3, M+1).
ER  -