On the global operator and Fueter mapping theorem for slice polyanalytic functions

Abstract

A. In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic se ing. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.

On the polyanalytic short-time Fourier transform in the quaternionic setting

“…e study of the QSTFT with respect to the weighted Hermite functions as windows is related to the theory of slice polynanalytic functions of a quaternionic variable. Recently, this topic has been intensively investigated, see [7,8,9,6]. e idea of the paper is to fix the following property…”

Generalized Appell polynomials and Fueter-Bargmann transforms in the polyanalytic setting

“…is paper proposes a generalization to the polyanalytic se ing of both Appell polynomials and the Bargmann-Fock-Fueter transform, studied in [10,11] and [17] respectively. A fundamental tool to perform the computations is the polyanalytic Fueter map, introduced in [3]. One of the main differences with respect to the classic Fueter theorem, see [24], is that in the polyanalytic se ing there are two Fueter mappings.…”

“…e second one, denoted by τ n+1 , maps slice polyanalytic functions of order n + 1 to Fueter regular ones. We note that there exists a relation between these two polyanalytic Fueter maps that can be expressed in terms of a suitable power of the Cauchy-Fueter operator, see [3]. In Section 3 we introduce a new family of polynomials in the quaternionic se ing (1.1) M k,s (q, q) := x k 0 Q s (q, q), k = 0, ..., n, s ≥ 0, where Q s (q, q) = s j=0 2(s−j+1) (s+1)(s+2) q s−j q j and x 0 is the real part of the quaternion q ∈ H. ese polynomials are obtained by applying the polyanalytic Fueter map C n+1 to a slice polyanalytic function of order n + 1, based on the series expansion theorem.…”

“…We note that one can extend the classical theory of holomorphic functions to higher order using the notion of polyanalytic functions, see [6]. Moreover, in the last years the notion of slice hyperholomorphic functions was extended also to this polyanalytic se ing, see [1,2,3,7]. We briefly recall this notion here and related results that will be needed in the sequel…”

“…Remark 2.11. We note that two Fueter mapping theorems were proved in the polyanalytic se ing, see [3]. is will allow to introduce the so-called polyanalytic Fueter mappings C n+1 and τ n+1 which will be studied and investigated in the next sections.…”