H-harmonic Reproducing Kernels on the Ball

We consider the Szegő reproducing kernel associated with the space of H-harmonic functions on the unit ball in n-dimensional space, i.e. functions that are characterized by being annihilated by the hyperbolic Laplacian. This paper derives an explicit series expansion for the reproducing kernel in terms of a triple hypergeometric function introduced of Exton. Moreover, we demonstrate that the Szegő kernel admits a representation as a finite sum of hypergeometric functions. We further show that the Szegő kernel, for linearly dependent arguments, can be expressed in terms of the first Appell hypergeometric function. In addition we provide a series expansion for the weighted Bergman kernels.