n-Regular functions in quaternionic analysis

In this paper, we study left and right [Formula: see text]-regular functions that originally were introduced in [I. Frenkel M. Libine, Quaternionic analysis, representation theory physics II, accepted Adv. Theor. Math. Phys]. When text], these are the usual quaternionic regular functions. We show satisfy most of properties functions, including conformal invariance under fractional linear transformations by group Cauchy–Fueter type reproducing formulas. Arguably, formulas for analogues Cauchy’s integral formula text]th-order pole text] also find two expansions kernel terms certain basis give an analogue Laurent series expansion construct invariant pairing between describe irreducible representations associated to spaces its Lie algebra.