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]

We consider a quaternionic analogue of the univariate complex Hermite polynomials and study some of their analytic properties in some detail. We obtain their integral representation as well as the operational formulas of exponential and Burchnall types they obey. We show that they form an orthogonal basis of the slice Hilbert space L2false(LI;e−false|q|2dλIfalse) of all quaternionic‐valued functions defined the whole quaternions space and subject to norm boundedness with respect to the Gaussian measure on a given slice as well as of the full left quaternionic Hilbert space L2false(double-struckH;e−false|q|2dλfalse) of square integrable functions on quaternions with respect to the Gaussian measure on the whole double-struckH≡R4. We also provide different types of generating functions. Remarkable identities, including quadratic recurrence formulas of Nielsen type, are also derived.

ABSTRACT. We prove that the complex Hermite polynomials H m,n on the complex plane C can be realized as the Fourier-Wigner transform V of the well-known real Hermite functions h n on real line R. This reduces considerably the Wong's proof [19, Chapter 21] giving the explicit expression of V(h m , h n ) in terms of the Laguerre polynomials. Moreover, we derive a new generating function for the H m,n as well as some new integral identities.

No abstract

We introduce new functional spaces that generalize the weighted Bergman and Dirichlet spaces on the disk D(0, R) in the complex plane and the Bargmann-Fock spaces on the whole complex plane. We give a complete description of the considered spaces. Mainly, we are interested in giving explicit formulas for their reproducing kernel functions and their asymptotic behavior as R goes to infinity.