  We study a new class of functions that arise naturally in quaternionic
analysis, we call them "quasi regular functions". Like the well-known
quaternionic regular functions, these functions provide representations of the
quaternionic conformal group. However, unlike the regular functions, the quasi
regular ones do not admit an invariant unitary structure but rather a
pseudounitary equivalent. The reproducing kernels of these functions have an
especially simple form: (Z-W)^{-1}. We describe the K-type bases of quasi
regular functions and derive the reproducing kernel expansions. We also show
that the restrictions of the irreducible representations formed from the quasi
regular functions to the Poincare group have three irreducible components.
  Our interest in the quasi regular functions arises from an application to the
study of conformal-invariant algebras of quaternionic functions. We also
introduce a factorization of certain intertwining operators between tensor
products of spaces of quaternionic functions. This factorization is obtained
using fermionic Fock spaces constructed from the quasi regular functions.
