Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 079, 20 pages      arXiv:1508.06689
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

A Hypergeometric Integral with Applications to the Fundamental Solution of Laplace's Equation on Hyperspheres

Richard Chapling
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England

Received November 23, 2015, in final form August 04, 2016; Published online August 10, 2016

We consider Poisson's equation on the $n$-dimensional sphere in the situation where the inhomogeneous term has zero integral. Using a number of classical and modern hypergeometric identities, we integrate this equation to produce the form of the fundamental solutions for any number of dimensions in terms of generalised hypergeometric functions, with different closed forms for even and odd-dimensional cases.

Key words: hyperspherical geometry; fundamental solution; Laplace's equation; separation of variables; hypergeometric functions.

pdf (437 kb)   tex (23 kb)


  1. Aubin T., Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
  2. Cohl H.S., Fundamental solution of Laplace's equation in hyperspherical geometry, SIGMA 7 (2011), 108, 14 pages, arXiv:1108.3679.
  3. Cohl H.S., Kalnins E.G., Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry, J. Phys. A: Math. Theor. 45 (2012), 145206, 32 pages, arXiv:1105.0386.
  4. Cohl H.S., Palmer R.M., Fourier and Gegenbauer expansions for a fundamental solution of Laplace's equation in hyperspherical geometry, SIGMA 11 (2015), 015, 23 pages, arXiv:1405.4847.
  5. Courant R., Hilbert D., Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953.
  6. Dutka J., The early history of the hypergeometric function, Arch. Hist. Exact Sci. 31 (1984), 15-34.
  7. Euler L., Specimen transformationis singularis serierum [E710], Nova Acta Academiae Scientarum Imperialis Petropolitinae 12 (1801), 58-70, reprinted in Opera Omnia, Ser. 1, Vol. 16-2, Birkhäuser, 1935, 41-55.
  8. Gauss C.F., Disquisitiones generales circa seriem infinitam $1 + \frac {\alpha \beta} {1 \cdot \gamma}x + \frac {\alpha (\alpha+1) \beta (\beta+1)} {1 \cdot 2 \cdot \gamma (\gamma+1)} xx + \text{etc.}$, Commentationes societatis regiae scientiarum Gottingensis recentiores 2 (1813), 1-46, reprinted in Gesammelte Werke, Bd. 3, K. Gesellschaft der Wissenschaften, Göttingen, 1876, 123-163, 207-229.
  9. Iliev B.Z., Handbook of normal frames and coordinates, Progress in Mathematical Physics, Vol. 42, Birkhäuser Verlag, Basel, 2006.
  10. Lee J.M., Introduction to smooth manifolds, Graduate Texts in Mathematics, Vol. 218, 2nd ed., Springer, New York, 2013.
  11. Pringsheim A., Faber G., Molk J., Analyse Algébrique, in Encyclopédie des sciences mathématiques, Tome II, Vol. II, Gauthier-Villars, Paris, 1911.
  12. Rainville E.D., The contiguous function relations for ${}_pF_q$ with applications to Bateman's $J_n^{u,v}$ and Rice's $H_n(\zeta,p,v)$, Bull. Amer. Math. Soc. 51 (1945), 714-723.
  13. Szmytkowski R., Closed forms of the Green's function and the generalized Green's function for the Helmholtz operator on the $N$-dimensional unit sphere, J. Phys. A: Math. Theor. 40 (2007), 995-1009.
  14. Vidūnas R., Contiguous relations of hypergeometric series, J. Comput. Appl. Math. 153 (2003), 507-519, math.CA/0109222.
  15. Vorsselmann de Heer P.O.C., Specimen Inaugurale De Fractionibus Continuis, Altheer, 1833, available at
  16. Yamasuge H., Maximum principle for harmonic functions in Riemannian manifolds, J. Inst. Polytech. Osaka City Univ. Ser. A. 8 (1957), 35-38.

Previous article  Next article   Contents of Volume 12 (2016)