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.

中文翻译：

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四元离子分析中的正则函数

在本文中，我们研究了 left 和 right [公式：见正文]——最初在 [I. Frenkel 和 M. Libine，四元数分析、表示论和物理学 II，在 Adv. 中接受。理论。数学。物理]。当【公式：见正文】时，这些函数就是通常的四元左右正则函数。我们证明了[公式：见正文]-正则函数满足通常正则函数的大部分性质，包括保形群的分数线性变换下的保形不变性和 Cauchy-Fueter 型再现公式。可以说，[公式：参见文本]-正则函数的这些 Cauchy–Fueter 类型再现公式是 [公式：参见文本]三阶极点的柯西积分公式的四元数类似物 [公式：