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Computing q+1Fq via rational conformal maps

  • John Lopez
  • Richard Mikael Slevinsky
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Abstract

It is well-known that certain Möbius (linear fractional) transformations provide analytic continuation of the Gauss hypergeometric function. These transformations work by exchanging the roles of the three regular singular points and $0$, $1$, and ∞ of the Gauss hypergeometric differential equation. Not one of these transformations, however, brings the points e±iπ/3 closer to the origin, and convergence in a neighbourhood of these is also troublesome. In this work, we produce a Maclaurin series in optimal complexity after rational conformal mapping that induces pointwise convergence to any q+1Fq throughout the cut complex plane ℂ∖[1,∞). We show that for the optimal parameter values this series converges faster than an approximate mapped Chebyshev series obtained by the ultraspherical spectral method.

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Cite

BibTeXRIS
BibTeX
@misc{lopez2026computing,
  title = {{Computing $_q+1F_q$ via rational conformal maps}},
  author = {John Lopez and Richard Mikael Slevinsky},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35577},
  eprint = {2610.35577},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35577},
}
RIS
TY  - UNPB
TI  - Computing _q+1F_q via rational conformal maps
AU  - John Lopez
AU  - Richard Mikael Slevinsky
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35577
UR  - https://arxiv.org/abs/2610.35577
AB  - It is well-known that certain Möbius (linear fractional) transformations provide analytic continuation of the Gauss hypergeometric function. These transformations work by exchanging the roles of the three regular singular points and $0$, $1$, and ∞ of the Gauss hypergeometric differential equation. Not one of these transformations, however, brings the points e^±iπ/3 closer to the origin, and convergence in a neighbourhood of these is also troublesome. In this work, we produce a Maclaurin series in optimal complexity after rational conformal mapping that induces pointwise convergence to any _q+1F_q throughout the cut complex plane ℂ∖[1,∞). We show that for the optimal parameter values this series converges faster than an approximate mapped Chebyshev series obtained by the ultraspherical spectral method.
ER  -