We introduce the notion of rationality for hyperholomorphic functions (functions in the kernel of the Cauchy-Fueter operator). Following the case of one complex variable, we give three equivalent definitions: the first in terms of Cauchy-Kovalevskaya quotients of polynomials, the second in terms of realizations and the third in terms of backward-shift invariance. Also introduced and studied are the counterparts of the Arveson space and Blaschke factors.
D. Alpay, M. Shapiro and D. Volok. Rational hyperholomorphic functions in R4. Journal of Functional Analysis, vol. 221 (2005) pp. 122-149.
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