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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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
Keywords
- Poisson kernel
- Hyperbolic upper half-space
- Poincaré unit ball
- Associated Legendre function
- Special functions