  We develop further quaternionic analysis introducing left and right doubly
regular functions. We derive Cauchy-Fueter type formulas for these doubly
regular functions that can be regarded as another counterpart of Cauchy's
integral formula for the second order pole, in addition to the one studied in
the first paper with the same title. We also realize the doubly regular
functions as a subspace of the quaternionic-valued functions satisfying a
Euclidean version of Maxwell's equations for the electromagnetic field.
  Then we return to the study of the original quaternionic analogue of Cauchy's
second order pole formula and its relation to the polarization of vacuum. We
find the decomposition of the space of quaternionic-valued functions into
irreducible components that include the spaces of doubly left and right regular
functions. Using this decomposition, we show that a regularization of the
vacuum polarization diagram is achieved by subtracting the component
corresponding to the one-dimensional subrepresentation of the conformal group.
After the regularization, the vacuum polarization diagram is identified with a
certain second order differential operator which yields a quaternionic version
of Maxwell equations.
  Next, we introduce two types of quaternionic algebras consisting of spaces of
scalar-valued and quaternionic-valued functions. We emphasize that these
algebra structures are invariant under the action of the conformal Lie algebra.
This uses techniques from our study of the vacuum polarization diagram. These
algebras are not associative, but we can define an infinite family of
n-multiplications, and we conjecture that they have structures of weak cyclic
A-infinity algebras. We also conjecture the relation between the multiplication
operations of the scalar and non-scalar quaternionic algebras with the n-photon
Feynman diagrams in the scalar and ordinary conformal QED.
