Generalized quaternionic Bargmann-Fock spaces and associated Segal-Bargmann transforms

S-Polyregular Bargmann Spaces

“…1,n follows in a similar way. It is also an immediate consequence of (19). The decomposition (19) readily follows since for given f ∈ SR 2 1,n , we have…”

“…Motivated by the works [2,3,10,29,37] studying and characterizing the polyanaliticity in the complex setting as well as by Brackx' works [13,14] studying the kmonogenic functions with respect to the Fueter operator, our aim in [19] was the study of possible generalizations of F 2 slice and its associated Segal-Bargmann transform to the context of slice n-polyregular (SR n ) functions with respect to the slice The research work of A.G. was partially supported by a grant from the Simons Foundation.…”

“…Such polynomials are the quaternionic analogues of the polyanalytic Hermite polynomials H m,n (z, z) ( [28,31]) that play a crucial role in studying some basic properties of polyanalytic functions [3]. The present paper is in fact an improved version of [19]. We consider two kinds of such generalizations.…”

Polyregularity of the dot product of slice regular functions

“…The first generalization in the quaternionic setting is due to Brackx [8,9] who defined the k-monogenic functions to be the elements of Ker(D) k+1 , D being the Füter operator. Its analog for the (left) slice derivative ∂ I is recently introduced in [11] (see also [12,7,4]). Thus, the solution of the generalized Cauchy-Riemann equations ∂ I n+1 f | I = 0; I 2 = −1, leads to the space SR n of S-polyregular functions of level n (or order n + 1), i.e., such that the restriction to any slice C I is polyanalytic of level n. Some basic properties of this new class of S-polyregular functions have been established…”