Bivariate Poly-analytic Hermite Polynomials
- PDF / 487,668 Bytes
- 21 Pages / 439.37 x 666.142 pts Page_size
- 56 Downloads / 197 Views
Results in Mathematics
Bivariate Poly-analytic Hermite Polynomials Allal Ghanmi
and Khalil Lamsaf
Abstract. A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier–Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical Hilbert space on the two-complex space with respect to the Gaussian measure. Their basic properties are discussed, such as their three term recurrence relations, operational realizations and differential equations (Bochner’s property) they obey. Different generating functions of exponential type are obtained. Integral and exponential operational representations are also derived. Some applications in the context of integral transforms and the concrete spectral theory of specific magnetic Laplacians are discussed. Mathematics Subject Classification. 33C50, 44A15, 44A20. Keywords. Bivariate poly-analytic Hermite polynomials, Fourier–Wigner transform, orthogonality, generating functions, integral representation, operational representation, magnetic Laplacian.
1. Introduction The so-called univariate (poly-analytic) complex Hermite polynomials (UHCP), denoted Hm,n (z, z), constitute an orthogonal basis of the classical Hilbert space 2 on the complex plane with respect to the Gaussian measure e−|z| dxdy. They were introduced by Itˆ o [17] in the framework of complex Markov process and turned out to be useful in many different contexts. In fact, they have been used as a basic tool in the study of, for instance, the nonlinear analysis of traveling wave tube amplifiers [5], the spectral theory of some second order differential operators [12,19,27], the study of some special integral transforms [7,16], coherent states theory [3,4], combinatory [14,15] and signal processing [8,22]. For their basic properties and applications, one can refer to [7,9,11,12,14]. 0123456789().: V,-vol
3
Page 2 of 21
A. Ghanmi and K. Lamsaf
Results Math
Bivariate complex polynomials of Hermite type can be defined in many different ways. The natural ones consist of considering the tensor product Hm (z)Hn (w) of the univariate holomorphic Hermite polynomials Hm (z) or also by replacing z in Hm,n (z, z) by the variable w, leading to the two-variable holomorphic Hermite polynomials Hm,n (z, w) considered in [14]. A systematic study of their analytic properties is presented in [13]. See also [18] for a quite variant class in three variables Hm,n (z, w, u). The u variable can be seen as a physical parameter that interprets time or magnitude of a magnetic field [7,12]. The tensor product Hm,n (z, z)Hm ,n (w, w) gives rise to another class of bivariate poly-analytic Hermite polynomials. In the present paper, we introduce a nontrivial class of bivariate (polyanalytic) complex orthogonal polynomials. They are not a standard tensor product of the UHCP, but they are with special composition operators. More precisely, following the same scheme giving rise to the UCHP from the real Hermite polynomials via a like-binomia
Data Loading...
