The Mittag Leffler reproducing kernel Hilbert spaces of entire and analytic functions

This paper investigates the function theoretic properties of two reproducing kernel functions based on the Mittag-Leffler function that are related through a composition. Both spaces provide one parameter generalizations of the traditional Bargmann-Fock space. In particular, the Mittag-Leffler space of entire functions yields many similar properties to the Bargmann-Fock space, and several results are demonstrated involving zero sets and growth rates. The second generalization, the Mittag-Leffler space of the slitted plane, is a reproducing kernel Hilbert space (RKHS) of functions for which Caputo fractional differentiation and multiplication by zq (for q>0) are densely defined adjoints of one another.