Hypergeometric reproducing kernels and analytic continuation from a half-line

Indefinite inner product spaces of entire functions and functions analytic inside a disk are considered and their completeness studied. Spaces induced by the rotation invariant reproducing kernels in the form of the generalized hypergeometric function are completely characterized. A particular space generated by the modified Bessel function kernel is utilized to derive an analytic continuation formula for functions on R+ using the best approximation theorem of Saitoh. As a by-product several new area integrals involving Bessel functions are explicitly evaluated.