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 both the slice and the full Hilbert spaces on the quaternions with respect to the Gaussian measure. We also provide different types of generating functions. Remarkably identities, including quadratic recurrence formulas of Nielsen type are derived.
In recent years, the study of slice monogenic functions has attracted more and more attention in the literature. In this paper, an extension of the well-known Dirac operator is defined which allows to establish the Lie superalgebra structure behind the theory of slice monogenic functions. Subsequently, an inner product is defined corresponding to this slice Dirac operator and its polynomial null-solutions are determined. Finally, analogues of the Hermite polynomials and Hermite functions are constructed in this context and their properties are studied.
We introduce two classes of $(p,q)$-Itô--Hermite polynomials, the post-quantum analogs of the $q$-Itô--Hermite polynomials introduced recently by Ismail and Zhang. We study their basic properties such as their operational formulas of Rodrigues type, the corresponding raising and lowering operators as well as their generating functions and the $(p,p$-differential equations they obey.
We study the logarihtnmic asymptotic of multiple orthogonal polynomials arising in a mixed type Hermite-Padé approximation problem associated with the rational perturbation of a Nikishin system of functions. The formulas obtained allow to give exact estimates of the rate of convergence of the corresponding Hermite-Padé approximants.
We study the orthogonal complement of the Hilbert subspace considered by by van Eijndhoven and Meyers in [J. Math. Anal. Appl. 146 (1990), no. 1, 89--98} and associated to holomorphic Hermite polynomials. A polyanalytic orthonormal basis is given and the explicit expressions of the corresponding reproducing kernel functions and Segal--Bargmann integral transforms are provided.
We consider the Mellin transforms of certain generalized Hermite functions based upon certain generalized Hermite polynomials, characterized by a parameter $μ>-1/2$. We show that the transforms have polynomial factors whose zeros lie all on the critical line. The polynomials with zeros only on the critical line are identified in terms of certain $_2F_1(2)$ hypergeometric functions, being certain scaled and shifted Meixner-Pollaczek polynomials. Other results of special function theory are presented.
We propose an algorithm for producing Hermite-Padé polynomials of type I for an arbitrary tuple of $m+1$ formal power series $[f_0,\dots,f_m]$, $m\geq1$, about $z=0$ ($f_j\in{\mathbb C}[[z]]$) under the assumption that the series have a certain (`general position') nondegeneracy property. This algorithm is a straightforward extension of the classical Viskovatov algorithm for construction of Padé polynomials (for $m=1$ our algorithm coincides with the Viskovatov algorithm). The algorithm proposed here is based on a recurrence relation and has the feature that all the Hermite-Padé polynomials co…
In this paper we use a set of partial differential equations to prove an expansion theorem for multiple complex Hermite polynomials. This expansion theorem allows us to develop a systematic and completely new approach to the complex Hermite polynomials. Using this expansion, we derive the Poisson Kernel, the Nielsen type formula, the addition formula for the complex Hermite polynomials with ease. A multilinear generating function for the complex Hermite polynomials is proved.
In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class ${\mathscr L}$. The conjectures are based in part on the numerical experiments, made recently by the authors in [26] and [27].
We discuss a new approach to realization of the well-known Weierstrass's programme on efficient continuation of an analytic element corresponding to a~multivalued analytic function with finite number of branch points. Our approach is based on the use of Hermite--Padé polynomials.
In this paper we present a generalization of the classical Hermite polynomials to the framework of Clifford-Dunkl operators. Several basic properties, such as orthogonality relations, recurrence formulae and associated differential equations, are established. Finally, an orthonormal basis for the Hilbert modules arising from the corresponding weight measures is studied.
A new approach to the problem of the zero distribution of Hermite-Padé polynomials of type I for a pair of functions $f_1,f_2$ forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field, which is posed on a two-sheeted Riemann surface.
Let $[f_0,\dots,f_m]$ be a tuple of series in nonnegative powers of $1/z$, $f_j(\infty)\neq0$. It is supposed that the tuple is in "general position". We give a construction of type I and type II Hermite--Padé polynomials to the given tuple of degrees $\leq{n}$ and $\leq{mn}$ respectively and the corresponding $(m+1)$-multi-indexes with the following property. Let $M_1(z)$ and $M_2(z)$ be two $(m+1)\times(m+1)$ polynomial matrices, $M_1(z),M_2(z)\in\operatorname{GL}(m+1,\mathbb C[z])$, generated by type I and type II Hermite--Padé polynomials respectively. Then we have $M_1(z)M_2(z)\equiv I_{m…
We introduce and analyze some numerical results obtained by the authors experimentally. These experiments are related to the well known problem about the distribution of the zeros of Hermite--Padé polynomials for a collection of three functions $[f_0 \equiv 1,f_1,f_2]$. The numerical results refer to two cases: a pair of functions $f_1,f_2$ forms an Angelesco system and a pair of functions $f_1=f,f_2=f^2$ forms a (generalized) Nikishin system. The authors hope that the obtained numerical results will set up a new conjectures about the limiting distribution of the zeros of Hermite--Padé polynom…
In this paper are discussed the results of new numerical experiments on zero distribution of type I Hermite-Padé polynomials of order $n=200$ for three different collections of three functions $[1,f_1,f_2]$. These results are obtained by the authors numerically and do not match any of the theoretical results that were proven so far. We consider three simple cases of multivalued analytic functions $f_1$ and $f_2$, with separated pairs of branch points belonging to the real line. In the first case both functions have two logarithmic branch points, in the second case they both have branch points …
We consider the problem of zero distribution of the first kind Hermite--Padé polynomials associated with a vector function $\vec f = (f_1, \dots, f_s)$ whose components $f_k$ are functions with a finite number of branch points in plane. We assume that branch sets of component functions are well enough separated (which constitute the Angelesco case). Under this condition we prove a theorem on limit zero distribution for such polynomials. The limit measures are defined in terms of a known vector equilibrium problem. Proof of the theorem is based on the methods developed by H.~Stahl, A.~A.~Goncha…
We present a brief introduction to the theory of multiple orthogonal polynomials on the basis of known results for an important class of measures known as Nikishin systems. For type I and type II multiple orthogonal polynomials with respect to such systems of measures, we describe some of their most relevant properties regarding location and distribution of zeros as well as their weak and ratio asymptotic behavior.
We provide a theoretical study of a new family of orthogonal functions on the punctured complex plane solving the eigenvalue problems for some magnetic Laplacian perturbed by a singular vector potential with zero magnetic field modeling the Aharonov-Bohm effect. The functions are defined by their $β$-modified Rodrigues type formula and extend the poly-analytic Itô--Hermite polynomials to the poly-meromorphic setting. Mainly, we derive their different operational representations and give their explicit expressions in terms of special functions. Different generating functions and integral repres…
We prove expressions for the inequalities in Hermite's theorem which are conditions for a real polynomial to have real zeros. These expressions generalize the discriminant of a quadratic polynomial and the expression of J. Marík for a cubic polynomial. We show that the $(k+1)$-th minor of the Hermite matrix associated a polynomial $p(x)$ is equal to the $k$-th minor of another matrix we call $E(n)$ times $n^{k-1}$ and a simple integer. To prove this equivalence, we prove generalizations of the discriminant of a polynomial and analyze certain labeled directed graphs. To define this matrix $E(n)…
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 representatio…
A class of integral transforms, on the planar Gaussian Hilbert space with range in the weighted Bergman space on the bi-disk, is defined as the dual transforms of the 2d fractional Fourier transform associated with the Mehler function for Itô--Hermite polynomials. Some spectral properties of these transforms are investigated. Namely, we study their boundedness and identify their null spaces as well as their ranges. Such identification depends on the zeros set of Itô--Hermite polynomials. Moreover, the explicit expressions of their singular values are given and compactness and membership in p-S…
