In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted as $\{\mathbb{H}_{n}(x;q)\}_{n\geq 0}$, which are orthogonal with respect to the following non-standard inner product involving q-differences: \begin{equation*} \langle p,q\rangle_{λ}=\int_{-1}^{1}f\left( x\right) g\left(x\right) (qx,-qx;q)_{\infty }d_{q}(x)+λ\,(\mathscr{D}_{q}^{j}f)(α)(\mathscr{D}_{q}^{j}g)(α), \end{equation*} where $α\in \mathbb{R}\backslash (-1,1)$, $λ$ belongs to the set of positive real numbers, $\mathscr{D}_{q}^{j}$ denotes the $j$-th $q $-discrete analogue of the derivative operator, and $(qx,-qx;q)_{\infty}d_{q}(x)$ denotes the orthogonality weight with its points of increase in a geometric progression. We proceed to obtain the hypergeometric representation of $\mathbb{H}_{n}(x;q)$ and explicit expressions for the corresponding ladder operators. From the latter, we obtain a novel kind of three-term recurrence formula with rational coefficients associated with these polynomial family. Moreover, for certain real values of $α$, we present some results concerning the location of the zeros of $\mathbb{H}_n(x;q)$ and we perform a comprehensive analysis of their asymptotic behavior as the parameter $λ$ varies from zero to infinity.
The q-Hermite I-Sobolev type polynomials of higher order are consider for their study. Their hypergeometric representation is provided together with further useful properties such as several structure relations which give rise to a three-term recurrence relation of their elements. Two different q-difference equations satisfied by the q-Hermite I-Sobolev type polynomials of higher order are also established.
We extend a randomisation method, introduced by Shiffman-Zelditch and developed by Burq-Lebeau on compact manifolds for the Laplace operator, to the case of $\mathbb{R}^d$ with the harmonic oscillator. We construct measures, thanks to probability laws which satisfy the concentration of measure property, on the support of which we prove optimal weighted Sobolev estimates on $\mathbb{R}^d$. This construction relies on accurate estimates on the spectral function in a non-compact configuration space. As an application, we show that there exists a basis of Hermite functions with good decay properti…
In this paper we find new equivalent norms in $L^p(\mathbb{R}^n,\mathbb{B})$ by using multivariate Littlewood-Paley functions associated with Poisson semigroup for the Hermite operator, provided that $\mathbb{B}$ is a UMD Banach space with the property ($α$). We make use of $γ$-radonifying operators to get new equivalent norms that allow us to obtain $L^p(\mathbb{R}^n,\mathbb{B})$-boundedness properties for (vector valued) multivariate spectral multipliers for Hermite operators. As application of this Hermite multiplier theorem we prove that the Banach valued Hermite Sobolev and potential spac…
In this paper, Hermite polynomials related to quantum systems with orthogonal O(m)-symmetry, finite reflection group symmetry G < O(m), symplectic symmetry Sp(2n) and superspace symmetry O(m) x Sp(2n) are considered. After an overview of the results for O(m) and G, the orthogonality of the Hermite polynomials related to Sp(2n) is obtained with respect to the Berezin integral. As a consequence, an extension of the Mehler formula for the classical Hermite polynomials to Grassmann algebras is proven. Next, Hermite polynomials in a full superspace with O(m) x Sp(2n) symmetry are considered. It is …
We present a new family $\left\{ S_{n}(x;q)\right\} _{n\geq 0}$ of monic polynomials in $x$, orthogonal with respect to a Sobolev-type inner product related to the $q$-Hermite I orthogonal polynomials, involving a first-order $q$-derivative on a mass-point $α\in \mathbb{R}$ located out of the corresponding orthogonality interval $[-1,1]$, for some fixed real number $q \in (0, 1)$. We present connection formulas, and the annihilation operator for this non-standard orthogonal polynomial family.
We study double integral representations of Christoffel-Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painleve' IV differential equation for each family.
A general family of matrix valued Hermite type orthogonal polynomials is introduced and studied in detail by deriving Pearson equations for the weight and matrix valued differential equations for these matrix polynomials. This is used to derive Rodrigues formulas, explicit formulas for the squared norm and to give an explicit expression of the matrix entries as well to derive a connection formula for the matrix polynomials of Hermite type. We derive matrix valued analogues of Burchnall formulas in operational form as well explicit expansions for the matrix valued Hermite type orthogonal polyno…
We use generating functions to express orthogonality relations in the form of $q$-beta integrals. The integrand of such a $q$-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal
In this paper we prove in a constructing way that exceptional Charlier, Meixner, Hermite and Laguerre polynomials satisfy higher order recurrence relations. Our conjecture is that the recurrence relations provided in this paper have minimal order.
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.
Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm-Liouville problem. Antonio Durán discovered a gap in the original proof of completeness for exceptional Hermite polynomials, that has propagated to analogous results for other exceptional families. In this paper we provide an alternative proof that follows essentially the same arguments, but provides a direct proof of the key lemma on which the completeness proof is based. This direct proof makes use of the theory of trivial monodromy potentials developed by Duistermaat an…
Using Casorati determinants of Charlier polynomials, we construct for each finite set $F$ of positive integers a sequence of polynomials $r_n^F$, $n\in σ_F$, which are eigenfunction of a second order difference operator, where $σ_F$ is an infinite set of nonnegative integers, $σ_F \varsubsetneq \NN$. For certain finite sets $F$ (we call them admissible sets), we prove that the polynomials $r_n^F$, $n\in σ_F$, are actually exceptional Charlier polynomials; that is, in addition, they are orthogonal and complete with respect to a positive measure. By passing to the limit, we transform the Casorat…
Let $μ$ be a finite positive Borel measure supported on R, $\LL[f] =xf''+(α+1-x)f'$ with $α>-1$, or $\LL[f] =\frac{1}{2}f''-xf'$, and $m$ a natural number. We study algebraic, analytic and asymptotic properties of the sequence of monic polynomials $\{Q_n\}_{n>m}$ orthogonal with respect to the operator $\LL$. We also provide a fluid dynamics model for the zeros of these polynomials.
We introduce a large class of Sobolev bi-orthogonal polynomial sequences arising from a $LU$-factorizable moment matrix and associated with a suitable measure matrix that characterizes the Sobolev bilinear form. A theory of deformations of Sobolev bilinear forms is also proposed. We consider both polynomial deformations and a class of transformations related to the action of linear operators on the entries of a given bilinear form. Transformation formulae among new and old polynomial sequences are determined. Finally, integrable hierarchies of evolution equations arising from the factorization…
A new recurrence relation for exceptional orthogonal polynomials is proposed, which holds for type 1, 2 and 3. As concrete examples, the recurrence relations are given for Xj-Hermite, Laguerre and Jacobi polynomials in j = 1,2 case.
We review properties of the $q-$Hermite polynomials and indicate their links with the Chebyshev, Rogers--Szegö, Al-Salam--Chihara, continuous $q-$% utraspherical polynomials. In particular we recall the connection coefficients between these families of polynomials. We also present some useful and important finite and infinite expansions involving polynomials of these families including symmetric and non-symmetric kernels. In the paper we collect scattered throughout literature useful but not widely known facts concerning these polynomials. It is based on 43 positions of predominantly recent li…
The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions -- the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical computations. In this paper we provide a partial solution to these problems. First, we construct an explicit basis of (non-orthogonal) eigenvectors of the discrete Fourier transform, thus extending the r…
We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux-Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2m, and they are indexed by the partitions λof m. We provide explicit expressions for their cor…
We study a family of the Laurent biorthogonal polynomials arising from the Hermite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients, explicit expression and the weight function for these polynomials are obtained. We construct also a new explicit example of the Szegö polynomials orthogonal on the unit circle. Relations with associated Legendre polynomials are considered.
On a general method for deriving a fourth-order differential equation satisfied by Laguerre-Hahn orthogonal polynomials with new results for the class 0 analogous to Hermite
University of Monastir（莫纳斯提尔大学）；Sorbonne Université（索邦大学）；Universidade do Porto（波尔图大学）
For every system $\{ p_n(z) \}_{n=0}^\infty$ of OPRL or OPUC, we construct Sobolev orthogonal polynomials $y_n(z)$, with explicit integral representations involving $p_n$. Two concrete families of Sobolev orthogonal polynomials (depending on an arbitrary number of complex parameters) which are generalized eigenvalues of a difference operator (in $n$) and generalized eigenvalues of a differential operator (in $n$) are given. Applications of a general connection between Sobolev orthogonal polynomials and orthogonal systems of functions in the direct sum of scalar $L^2_μ$ spaces are discussed.
Recently there has been a renewed interest in an extension of the notion of orthogonal polynomials known as multiple orthogonal polynomials. This notion comes from simultaneous rational approximation (Hermite-Pade approximation) of a system of several functions. We describe seven families of multiple orthogonal polynomials which have he same flavor as the very classical orthogonal polynomials of Jacobi, Laguerre and Hermite. We also mention some open research problems and some applications.
In this paper we characterize the Rogers q-Hermite polynomials as the only orthogonal polynomial set which is also ${\cal D}_q$-Appell where ${\cal D}_q $ is the Askey-Wilson finite difference operator.
We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket ($SG$). These orthogonal polynomials arise through the Gram-Schmidt orthogonalisation process applied on the set of monomials on $SG$ using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their $L^2$, $L^\infty$ and Sobolev norms, and study their asymptotic behaviour. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.
In the paper we study the multipliers of Fourier series in polynomials orthogonal in continuous-discrete Sobolev's spaces. Multiplier Theorem for Fourier-Sobolev series is obtained. This result is based on the representation of the Fejér kernel, on the construction of the "humpbacked majorant" and weighted estimates of maximal functions.
We investigate the global density of zeros of generalized Hermite orthogonal polynomials, subject to certain truncated conditions on its weight. We shall given explicitly the global density of zeros under some asymptotic conditions on the weight. Moreover we compute the asymptotic of the total energy of the equilibrium position of the system of $n$ movable unit charges in an external field determined by the weight of the generalized Hermite polynomials. We will see that for finite $n$ the energy is in direct relationship with the zeros of the orthogonal polynomials.
Properties of certain $q$-orthogonal polynomials are connected to the $q$-oscillator algebra. The Wall and $q$-Laguerre polynomials are shown to arise as matrix elements of $q$-exponentials of the generators in a representation of this algebra. A realization is presented where the continuous $q$-Hermite polynomials form a basis of the representation space. Various identities are interpreted within this model. In particular, the connection formula between the continuous big $q$-Hermite polynomials and the continuous $q$-Hermite polynomials is thus obtained, and two generating functions for thes…
We show that the only orthogonal polynomials with a generating function of the form $F(x z - αz^2)$ are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. For special $F$ for which this is the case, we then finish the classification of orthogonal polynomials with more general generating functions $F(x w(z) - R(z))$.
In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial resul…
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Laguerre-Sobolev bilinear form with mass point at zero. In particular we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that they are eigenfunctions of a differential operator (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrix which defines the discrete Laguerre-Sobolev bilinear form.
We investigate the asymptotic properties of orthogonal polynomials for a class of inner products including the discrete Sobolev inner products $\langle h,g \rangle = \int hg\, dμ+ \sum_{j=1}^m \sum_{i=0}^{N_j} M_{j,i} h^{(i)}(c_j) g^{(i)}(c_j)$, where $μ$ is a certain type of complex measure on the real line, and $c_j$ are complex numbers in the complement of $\supp(μ)$. The Sobolev orthogonal polynomials are compared with the orthogonal polynomials corresponding to the measure $μ$.
We introduce and develop a theory of orthogonality with respect to Sobolev inner products on the real line for sequences of functions with a tridiagonal, skew-Hermitian differentiation matrix. While a theory of such L2-orthogonal systems is well established, Sobolev orthogonality requires new concepts and their analysis. We characterise such systems completely as appropriately weighed Fourier transforms of orthogonal polynomials and present a number of illustrative examples, inclusive of a Sobolev-orthogonal system whose leading N coefficients can be computed in $\mathcal{O}(N \log N)$ operati…
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at $-1$ and/or $+1$. In particular, we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that when the Jacobi parameters $α$ and $β$ are nonnegative integers the Jacobi-Sobolev orthogonal polynomials are eigenfunctions of a differential operator of finite order (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of…
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 f…
Sobolev orthogonal polynomials have been studied extensively in the past 20 years. The research in this field has sprawled into several directions and generates a plethora of publications. This paper contains a survey of the main developments up to now. The goal is to identify main ideas and developments in the field, which hopefully will lend a structure to the mountainous publications and help future research.
We introduce bivariate versions of the continuous q-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous q-Hermite polynomials and the star product method of [Kolb and Yakimov, Adv. Math. 2020] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac-Moody type. We prove that the…
We investigate the uniform asymptotic of some Sobolev orthogonal polynomials. Three term recurrence relation is given, moreover we give a recurrence relation between the so-called Sobolev orthogonal polynomials and Freud orthogonal polynomials.
Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big $q…
Motivated by the Novikov equation and its peakon problem, we propose a new mixed type Hermite--Padé approximation whose unique solution is a sequence of polynomials constructed with the help of Pfaffians. These polynomials belong to the family of recently proposed partial-skew-orthogonal polynomials. The relevance of partial-skew-orthogonal polynomials is especially visible in the approximation problem germane to the Novikov peakon problem so that we obtain explicit inverse formulae in terms of Pfaffians by reformulating the inverse spectral problem for the Novikov multipeakons. Furthermore, w…
The bispectral anti-isomorphism is applied to differential operators involving elements of the stabilizer ring to produce explicit formulas for all difference operators having any of the Hermite exceptional orthogonal polynomials as eigenfunctions with eigenvalues that are polynomials in $x$.
A family of orthonormal polynomials on the unit ball $B^d$ of $\RR^d$ with respect to the inner product $$ < f,g > = \int_{B^d}Δ[(1-\|x\|^2) f(x)] Δ[(1-\|x\|) g(x)] dx, $$ where $Δ$ is the Laplace operator, is constructed explicitly.
In this paper, the structures to a family of biorthogonal polynomials that approximate to the Hermite and Generalized Laguerre polynomials are discussed respectively. Therefore, the asymptotic relation between several orthogonal polynomials and combinatorial polynomials are derived from the systems, which in turn verify the Askey scheme of hypergeometric orthogonal polynomials. As the applications of these properties, the asymptotic representations of the generalized Buchholz, Laguerre, Ultraspherical(Gegenbauer), Bernoulli, Euler, Meixner and Meixner-Pllaczekare polynomials are derived from t…
In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions.…
In this paper we study the following family of hypergeometric polynomials: $y_n(x) = \frac{ (-1)^ρ}{ n! } x^n {}_2 F_0(-n,ρ;-;-\frac{1}{x})$, depending on a parameter $ρ\in\mathbb{N}$. Differential equations of orders $ρ+1$ and $2$ for these polynomials are given. A recurrence relation for $y_n$ is derived as well. Polynomials $y_n$ are Sobolev orthogonal polynomials on the unit circle with an explicit matrix measure.
In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we are focused in the study of the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some new directions for a future research are formulated.
It is shown that if a non-zero function $f\in B_σ$ has infinitely many double zeros on the real axis, then there exists at least one pair of consecutive zeros whose distance apart is greater than $\dfracπστ^{1/4}$, $τ\approx5.0625$. A frame algorithm is provided for reconstructing a function $f\in B_σ$ from its nonuniform samples $\{f^{(j)}(x_i):j=0,1,\dots, k-1, i\in\mathbb{Z}\}$ with maximum gap condition, $\sup\limits_i(x_{i+1}-x_i)=δ<\dfrac{1}σc_k^{1/2k}$, where $c_k$ is a Wirtinger-Sobolev constant, using Hermite interpolation.
In the present paper, we deal mainly with arithmetic properties of Legendre polynomials by using their orthogonality property. We show that Legendre polynomials are proportional with Bernoulli, Euler, Hermite and Bernstein polynomials.
We analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which involves the outward normal derivatives on the sphere. Using their representation in terms of spherical harmonics, algebraic and analytic properties will be deduced. First, we deduce explicit connection formulas relating classical multivariate ball polynomials and our family of Sobolev orthogonal polynomials. Then explicit representations for the norms and the kernels will be obtained. Finally, the asymptotic behaviour of the corresponding Christoffel functions is studied.
The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these polynomials satisfy the differential equation of the second order obtained in this work or there is no differential equation of a finite order for these polynomials.
In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: ${}_2 F_2(-n,1;q,r;x)$ and ${}_3 F_2(-n,n-1+a+b,1;a,c;x)$ ($a,b,c,q,r>0$, $n=0,1,...$), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: $L y + λ_n D y = 0$, where $L,D$ are linear differential operators with polynomial coefficients not depending on $n$. For positive integer values of the parameters $r,c$ these polynomials are Sobolev orthogonal polyn…
Let the Sobolev-type inner product <f,g> = \int fg d mu_0+ int f' g' d mu_1 with mu_0 = w + M delta_c, mu_1= N delta_c where w is the Jacobi weight, c is either 1 or -1 and M, N >= 0. We obtain estimates and asymptotic properties on [-1,1] for the polynomials orthonormal with respect to <.,.> and their kernels. We also compare these polynomials with Jacobi orthonormal polynomials.
