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We develop some aspects of the theory of hyperholomorphic functions whose values are taken in a Banach algebra over a field—assumed to be the real or the complex numbers—and which contains the field. Notably, we consider Fueter expansions, Gleason’s problem, the theory of hyperholomorphic rational functions, modules of Fueter series, and related problems. Such a framework includes many familiar algebras as particular cases. The quaternions, the split quaternions, the Clifford algebras, the ternary algebra, and the Grassmann algebra are a few examples of them.


This is a pre-copy-editing, author-produced PDF of an article that was later accepted for publication in Israel Journal of Mathematics, volume 236, in 2020 following peer review. The final publication may differ and is available at Springer via

A free-to-read copy of the final published article is available here.

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