On a Slice Hyper-Meromorphic Bergman Space

We intend to introduce and investigate a new functional space on the quaternionic unit ball of slice hyper-meromorphic functions with unique pole at the origin. Mainly, we provide a concrete characterization of their elements and give the closed explicit expression of the associated reproducing kernel function. Moreover, we show that they are isometrically isomorphic to the configuration space on the positive real half line by means of an integral transform of Bargmann type. The closed formulas for this transform are given in two special cases.