We introduce two classes of right quaternionic Hilbert spaces in the context of slice polyregular functions, generalizing the so-called slice and full hyperholomorphic Bargmann spaces. Their basic properties are discussed, the explicit formulas of their reproducing kernels are given and associated Segal-Bargmann transforms are also introduced and studied. The spectral description as special subspaces of L 2 -eigenspaces of a second order differential operator involving the slice derivative is investigated. arXiv:1812.09129v3 [math.CV]

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We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different orthogonality identities. We establish their connection and rule in describing the L 2 -spectral theory of some special second order differential operators of Laplacian type acting on the L 2 -gaussian Hilbert space on the whole complex plane. We will also show their importance in the theory of the so-called rank-one automorphic functions on the complex plane. In fact, a variant subclass leads to an orthogonal basis of the corresponding L 2 -gaussian Hilbert space on the strip.

We introduce and discuss some basic properties of some integral transforms in the framework of specific functional Hilbert spaces, the holomorphic Bargmann-Fock spaces on C and C 2 and the sliced hyperholomorphic Bargmann-Fock space on H. The first one is a natural integral transform mapping isometrically the standard Hilbert space on the real line into the two-dimensional Bargmann-Fock space. It is obtained as composition of the one and two dimensional Segal-Bargmann transforms and reduces further to an extremely integral operator that looks like a composition operator of the one-dimensional Segal-Bargmann transform with a specific symbol. We study its basic properties, including the identification of its image and the determination of a like-left inverse defined on the whole two-dimensional Bargmann-Fock space. We also examine their combination with the Fourier transform which lead to special integral transforms connecting the two-dimensional Bargmann-Fock space and its analogue on the complex plane. We also investigate the relationship between special subspaces of the twodimensional Bargmann-Fock space and the slice-hyperholomorphic one on the quaternions by introducing appropriate integral transforms. We identify their image and their action on the reproducing kernel.2010 Mathematics Subject Classification. Primary 44A15; Secondary 32A17, 32A10.

Generating functions for the univariate complex Hermite polynomials (UCHP) are employed to introduce some non-trivial one and twodimensional integral transforms of Segal-Bargmann type in the framework of specific functional Hilbert spaces. The approach used is issued from the coherent states framework. Basic properties of these transforms are studied. Connection to some special known transforms like Fourier and Wigner transforms is established. The first transform is a two-dimensional Segal-Bargmann transform whose kernel is related the exponential generating function of the UCHP. The second one connects the so-called generalized Bargmann-Fock spaces (or also true-poly-Fock spaces in the terminology of Vasilevski) that are realized as the L 2 -eigenspaces of a specific magnetic Schrödinger operator. Its kernel function is related to a Mehler formula of the UCHP. 35A22; 44A15; 33C45; 42C05; 42A38; 32A10