Multidomain spectral method for the Gauss hypergeometric function

  • S. Crespo
  • M. Fasondini
  • C. KleinEmail author
  • N. Stoilov
  • C. Vallée
Original Paper


We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius’ method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line \(\mathbb {R}\cup {\infty }\), except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourier–ultraspherical spectral method. In cases where logarithms appear in the solution, a hybrid approach involving an analytical treatment of the logarithmic terms is applied. We show for several examples that machine precision can be reached for a wide class of parameters, but also discuss almost degenerate cases where this is not possible.


Hypergeometric function Singular differential equations Spectral methods 



We thank C. Lubich for helpful remarks.

Funding information

This work was partially supported by the PARI and FEDER programs in 2016 and 2017, by the ANR-FWF project ANuI by the isite BFC via the project NAANoD and by the Marie-Curie RISE network IPaDEGAN. M. Fasondini acknowledges financial support from the EPSRC grant EP/P026532/1.


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Authors and Affiliations

  1. 1.Institut de Mathématiques de BourgogneUniversité de Bourgogne-Franche-ComtéDijon CedexFrance
  2. 2.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUK
  3. 3.Univ. Rennes, Inria, CNRS, IRISA, Inserm, Empenn U1228RennesFrance

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