Mehler's formulas for the univariate complex Hermite polynomials and applications

We give two widest Mehler's formulas for the univariate complex Hermite polynomials $H_{m,n}^\nu$, by performing double summations involving the products $u^m H_{m,n}^\nu (z,\overline{z}) \overline{H_{m,n}^\nu (w,\overline{w})}$ and $u^m v^n H_{m,n}^\nu (z,\overline{z}) \overline{H_{m,n}^{\nu'} (w,\overline{w})}$. They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level $m$. The second Mehler's formula generalizes the one appearing as a particular case of the so-called Kibble-Slepian formula. The proofs, we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.Comment: 5 pages. To appear in Math. Methods Appl. Sc

On L2L^2-eigenfunctions of Twisted Laplacian on curved surfaces and suggested orthogonal polynomials

We show in a unified manner that the factorization method describes completely the $L^2$-eigenspaces associated to the discrete part of the spectrum of the twisted Laplacian on constant curvature Riemann surfaces. Subclasses of two variable orthogonal polynomials are then derived and arise by successive derivations of elementary complex valued functions depending on the geometry of the surface.Comment: This will appear in the Proceeding of ICOAA08 that will be published by the Journal of Operators and Matrices

Construction of concrete orthonormal basis for (L^2,\Gamma,\chi)-theta functions associated to discrete subgroups of rank one in (C,+)

Let \chi be a character on a discrete subgroup \Gamma of rank one of the additive group (C,+). We construct a complete orthonormal basis of the Hilbert space of (L^2,\Gamma,\chi)-theta functions. Furthermore, we show that it possesses a Hilbertian orthogonal decomposition involving the L^2-eigenspaces of the Landau operator \Delta_\nu; \nu>0, associated to the eigenvalues \nu m. For m=0, the associated L^2-eigenspace is the Hilbert subspace of entire (L^2,\Gamma,\chi)-theta functions. Corresponding orthonormal basis are constructed and the corresponding reproducing kernel can be expressed in terms of the generalized theta function of characteristic [\alpha,0].Comment: 17 page

An integral representation for Folland's fundamental solution of the sub-Laplacian on Heisenberg groups Hn\Bbb{H}^{n}

We prove that the Folland's fundamental solution for the sub-Laplacian on Heisenberg groups can also be derived form the resolvent kernel of this sub-Laplacian. This provides us with a new integral representation for this fundamental solution.Comment: 5 pages, mistakes corrected and a refrence adde

Bicomplex analogs of Segal-Bargmann and fractional Fourier transforms

We consider and discuss some basic properties of the bicomplex analog of the classical Bargmann space. The explicit expression of the integral operator connecting the complex and bicomplex Bargmann spaces is also given. The corresponding bicomplex Segal--Bargmann transform is introduced and studied as well. Its explicit expression as well as the one of its inverse are then used to introduce a class of two--parameter bicomplex Fourier transforms (bicomplex fractional Fourier transform). This approach is convenient in exploring some useful properties of this bicomplex fractional Fourier transform.Comment: 16 page

On a novel class of polyanalytic Hermite polynomials

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.Comment: 18 page

On the range of weighted planar Cauchy transform

We describe the range of of weighted Cauchy transform and its $k$-Bergman projection when action on weighted true poly-Bargmann spaces constituting an orthogonal Hilbertian decomposition of the Hilbert space of Gaussian functions on the complex plane.Comment: 6 pages Dedicated to the memory of Ahmed Intissar passed away in July 26, 2017

On a class of two-index real Hermite polynomials

We introduce a class of doubly indexed real Hermite polynomials and we deal with their related properties like the associated recurrence formulae, Runge's addition formula, generating function and Nielsen's identity.Comment: 6 page

Polyregularity of the dot product of slice regular functions

In this paper, we are concerned with the S-polyregularity the regular dot product of slice regular functions. We prove that the product of a slice regular function and right quaternionic polynomial function is a S-polyregular function and we determinate its exact order. The general case of the product of any two slice regular functions is also discussed. In fact, we provide sufficient and necessary conditions to the product of slice regular functions be a S-polyregular function. The obtained results are then extended to the product of S-polyregular functions and remain valid for a special dot product. As consequences we obtain linearization theorems for such S-polyregular products with respect to the slice regular functions

On dual transform of fractional Hankel transform

We deal with a class of one-parameter family of integral transforms of Bargmann type arising as dual transforms of fractional Hankel transform. Their ranges are identified to be special subspaces of the weighted hyperholomorphic left Hilbert spaces, generalizing the slice Bergman space of the second kind. Their reproducing kernel is given by closed expression involving the $\star$-regularization of Gauss hypergeometric function. We also discuss their basic properties such as their boundedness and we determinate their singular values. Moreover, we describe their compactness and membership in $p$-Schatten classes.Comment: 10 page