N-Regular functions in quaternionic analysis

Abstract

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