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Special function representations of the Poisson kernel on hyperbolic spaces

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Abstract

In this paper we compute explicit formulae for the Poisson kernels on the hyperbolic upper half-space \(\mathbf {H}^{n}\) and the Poincaré unit ball \(\mathbf {D}^{n}\). We first construct an associated Legendre function expression for eigenfunctions of the Laplacian and use superposition principle to get a solution for the Laplace equation on \(\mathbf {H}^{n}\). The Poisson kernel on \(\mathbf {D}^{n}\) is obtained from that on \(\mathbf {H}^{n}\) by letting the hyperbolic distance \(\rho =d(w,w')\) \((w,w'\in \mathbf {H}^{n})\) tend to infinity. These Poisson kernels, apart from being interesting in their own right lead to various identities that seem to be novel in the context of special functions.

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References

  1. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications Inc., New York, 1972)

    Google Scholar 

  2. H. Bateman, A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions II, Graduate Texts in Mathematics (Springer, New York, 1976)

    Google Scholar 

  3. Y.A. Brychkov, Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas (CRC Press, Boca Raton, 2008)

    Google Scholar 

  4. T. Byczkowski, P. Graczyk, A. Stós, Poisson kernels of half-spaces in real hyperbolic spaces. Revista Matemática Iberoamericana 23, 85–126 (2007)

    Article  Google Scholar 

  5. T. Byczkowski, J. Małecki, Poisson kernel and Green function of the ball in real hyperbolic spaces. Potential Anal. 27, 1–26 (2007)

    Article  Google Scholar 

  6. V. Cammarota, E. Orsingher, Hitting spheres on hyperbolic spaces. Theory Probab. Appl. 57, 419–443 (2013)

    Article  Google Scholar 

  7. I. Chavel, Eigenvalues in Riemannian Geometry (Academic Press, New York, 1984)

    Google Scholar 

  8. J. Dougall, A theorem of Sonine in Bessel functions, with two extensions to spherical harmonics. Proc. Edinb. Math. Soc. 37, 33–47 (1918)

    Article  Google Scholar 

  9. G.B. Folland, Spherical harmonic expansion of the Poisson-Szegő kernel for the ball. Proc. AMS 47, 401–408 (1975)

    Google Scholar 

  10. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products, 7th edn. (Academic Press Inc., New York, 2007)

    Google Scholar 

  11. S. Grellier, J.P. Otal, Bounded eigenfunctions in the real hyperbolic space. Int, Math. Res. Not. x, 3867–3897 (2005)

    Article  Google Scholar 

  12. M. Hashizume, A. Kowata, K. Minemura, K. Okamoto, An integral representation of an eigenfunction of the Laplacian on the Euclidean space. Hiroshima Math. J. 2, 535–545 (1972)

    Google Scholar 

  13. S. Helgason, Eigenspaces of the Laplacian: integral representations and irreducibility. J. Funct. Anal. 17, 328–353 (1974)

    Article  Google Scholar 

  14. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press, New York, 1979)

    Google Scholar 

  15. S. Helgason, Groups and Geometric Analysis: Radon Transforms, Invariant Differential Operators and Spherical functions (Academic Press, New York, 1984)

    Google Scholar 

  16. P. Jaming, Harmonic functions on the real hyperbolic ball I: boundary values and atomic decomposition of Hardy spaces. Coll. Math. 80, 63–82 (1999)

    Article  Google Scholar 

  17. K. Minemura, Eigenfunctions of the Laplacian on a real hyperbolic space. J. Math. Soc. Jpn. 27, 82–105 (1975)

    Article  Google Scholar 

  18. K. Minemura, Harmonic functions on real hyperbolic spaces. Hiroshima Math. J. 3, 121–151 (1973)

    Google Scholar 

  19. Z. Mouayn, Poisson integral representation of some eigenfunctions of Landau Hamiltonian on the hyperbolic disc. Bull. Belg. Math. Soc. 12, 249–257 (2005)

    Google Scholar 

  20. B. Muckenhoupt, E.M. Stein, Classical expansions and their relation to conjugate harmonic functions. Trans. AMS 118, 17–92 (1965)

    Article  Google Scholar 

  21. T. Sergo, Boundary Properties and Applications of the Differentiated Poisson Integral for Different Domains (Nova Science Publishers Inc, New York, 2009)

    Google Scholar 

  22. E. Symeonidis, The Poisson integral for a disc on the 2-sphere. Expo. Math. 17, 365–370 (1999)

    Google Scholar 

  23. E. Symeonidis, The Poisson integral for a disk in the hyperbolic plane. Expo. Math. 17, 239–244 (1999)

    Google Scholar 

  24. E. Symeonidis, The Poisson integral for a ball in spaces of constant curvature. Commun. Math. Univ. Carol. 44, 437–460 (2003)

    Google Scholar 

  25. E. Symeonidis, Das Poisson-Integral für Kugeln in Räumen konstanter Krümmung (Logos Verlag Berlin, Berlin, 2004)

    Google Scholar 

  26. A. Taheri, Function Spaces and Partial Differential Equations. I & II, Oxford Lecture Series in Mathematics and Its Applications, vol. 40-41 (OUP, Oxford, 2015)

    Book  Google Scholar 

Download references

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Correspondence to Richard Olu Awonusika.

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Awonusika, R.O. Special function representations of the Poisson kernel on hyperbolic spaces. J Math Chem 56, 825–849 (2018). https://doi.org/10.1007/s10910-017-0833-x

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  • DOI: https://doi.org/10.1007/s10910-017-0833-x

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