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This is a continuation of a series of papers [FL1, FL2, FL3], where wedevelop quaternionic analysis from the point of view of representation theoryof the conformal Lie group and its Lie algebra. In this paper we continue tostudy the quaternionic analogues of Cauchy's formula for the second order pole.These quaternionic analogues are closely related to regularization ofinfinities of vacuum polarization diagrams in four-dimensional quantum fieldtheory. In order to add some flexibility, especially when dealing with Cauchy'sformula for the second order pole, we introduce a one-parameter deformation ofquaternionic analysis. This deformation of quaternions preserves conformalinvariance and has a geometric realization as anti de Sitter space sittinginside the five-dimensional Euclidean space. We show that many results ofquaternionic analysis - including the Cauchy-Fueter formula - admit a simpleand canonical deformation. We conclude this paper with a deformation of thequaternionic analogues of Cauchy's formula for the second order pole.

Anti De Sitter Deformation of Quaternionic Analysis and the Second Order Pole