In this paper, we introduce the quaternionic slice polyanalytic functions and we prove some of their properties. Then, we apply the obtained results to begin the study of the quaternionic Fock and Bergman spaces in this new setting. In particular, we give explicit expressions of their reproducing kernels.Proposition 3.18. Let Ω be an axially symmetric domain and let f : Ω −→ H be a slice polyanalytic function. Assume that there exist J, K ∈ S, with J = K and U J , U K two subdomains of Ω + J and Ω + K respectively where Ω + J := Ω ∩ C + J and Ω + K := Ω ∩ C + K . If f = 0 on U J and U K , then f = 0 everywhere in Ω. Proof. Let f be a slice polyanalytic function on Ω such that f = 0 on U J and U K . Thus, since U J and U K are respectively subdomains of Ω + J and Ω + K . It follows, from the Splitting Lemma for slice polyanalytic functions combined with the classical complex analysis that f = 0 everywhere on Ω + J and Ω + K . Then, we just need to use the Representation Formula which allows to recover a slice function by its values on two semi-slices to complete the proof.

The Bargmann-Fock space of slice hyperholomorphic functions is recently introduced by Alpay, Colombo, Sabadini and Salomon. In this paper, we reconsider this space and present a direct proof of its independence of the slice. We also introduce a quaternionic analogue of the classical Segal-Bargmann transform and discuss some of its basic properties. The explicit expression of its inverse is obtained and the connection to the left one-dimensional quaternionic Fourier transform is given.

A. is paper deals with some special integral transforms of Bargmann-Fock type in the se ing of quaternionic valued slice hyperholomorphic and Cauchy-Fueter regular functions. e construction is based on the well-known Fueter mapping theorem. In particular, starting with the normalized Hermite functions we can construct an Appell system of quaternionic regular polynomials. e ranges of such integral transforms are quaternionic reproducing kernel Hilbert spaces of regular functions. New integral representations and generating functions in this quaternionic se ing are obtained in both the Fock and Bergman cases.Kamal Diki : Marie Sklodowska-Curie fellow of the Istituto Nazionale di Alta Matematica . 1 20 which is defined making use of the slice hyperholomorphic extension operator, i.e.e next result relates the slice Bergman kernel on the quaternionic half ball to the slice Bergman kernels in the case of the quaternionic unit ball and of the half space.is a right quaternionic reproducing kernel Hilbert space. Moreover, for all (q, r) ∈ B + × B + we have:where K B and K H + are, respectively, the slice Bergman kernels of the quaternionic unit ball and half space.Proof.e first assertion follows from the general theory. en, let us fix r ∈ B + such that r belongs to the slice C J with J ∈ S. en, we consider the function ψ r defined byClearly ψ r belongs to A Slice (B + ) since B + is contained in both B and H + and since by definition K B and K H + are the slice Bergman kernels of the quaternionic unit ball and half space. en, we only need to prove the reproducing kernel property. Indeed, let f ∈ A Slice (B + ). In particular, by the Spli ing Lemma we can write f Jus, by applying the results from the classical complex se ing we getSo, it follows that the function ψ r belongs and reproduces any element of the space A Slice (B + ) for any r ∈ B + . Hence, by the uniqueness of the reproducing kernel we getis completes the proof. e explicit expression of the slice Bergman kernel of the quaternionic half-ball is given by the following eorem 5.3. For all (q, r) ∈ B + × B + , we have: K B + (q, r) = (1 + q 2 ) [(1 − qr) * (q + r)] − * 2 (1 + r 2 ),where the * -product is taken with respect to the variable q. 21

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 setting. 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.

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.