It is stated and proved a characterization theorem for Laguerre-Hahn orthogonal polynomials on non-uniform lattices. This theorem proves the equivalence between the Riccati equation for the formal Stieltjes function, linear first-order difference relations for the orthogonal polynomials as well as for the associated polynomials of the first kind, and linear first-order difference relations for the functions of the second kind.
We construct new examples of Krall discrete orthogonal polynomials, i.e., orthogonal polynomials with respect to a measure which are also eigenfunctions of a higher order difference operator. The new examples include the orthogonal polynomials with respect to the measures obtained from the Meixner measure $ρ_{a,c}$ and Hahn measure $ρ_{a,b,N}$ by removing a finite number of their mass points when the parameter $c$ of the Meixner measure and $a$ or $b$ of the Hahn measure are positive integers. From the new Krall-Meixner families we construct new families of exceptional Meixner and Laguerre pol…
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）
On the orthogonality of q-classical polynomials of the Hahn class II
本文从R.阿尔瓦雷斯-诺达尔塞河Sevinik-Adıgüzel和H.进行Tajaneli，\textit{On the orthogonality of $q$-classical polynomials of the Hahn class I}。更具体地说，属于$\emptyset$ -Hermite-Laguerre/Jacobi，$\emptyset$ -Jacobi/Hermite-Laguerre，0-Laguerre/Jacobi-Bessel和0-Jacobi/Laguerre-Bessel情况下的$q$ -多项式的正交性质进行了研究，考虑到在初始文件中考虑的想法。特别地，建立了$q$ -Meixner多项式的一个新的正交关系.
Considering a differential operator of third order that does not increase the degree of polynomials, we analyse some properties of elements of the dual space of 2-orthogonal polynomial eigenfunctions. In two classes of such generic operator, we prove that a 2-orthogonal polynomial solution fulfils Hahn's property.
We first show how one can obtain Al-Salam--Chihara polynomials, continuous dual $q$-Hahn polynomials, and Askey--Wilson polynomials from the little $q$-Laguerre and the little $q$-Jacobi polynomials by using special transformations. This procedure is then extended to obtain multiple Askey--Wilson, multiple continuous dual $q$-Hahn, and multiple Al-Salam--Chihara polynomials from the multiple little $q$-Laguerre and the multiple little $q$-Jacobi polynomials.
In this paper Fourier transform of multivariate orthogonal polynomials on the simplex is presented. A new family of multivariate orthogonal functions is obtained by using the Parseval's identity and several recurrence relations are derived.
Morales-Mendoza et al. present in 2013 a new class of discrete orthogonal polynomials. They use these polynomials to design an unbiased FIR filter. In their paper they make the statement that a representation of the polynomials via hypergeometric functions is unknown. However Shakibaei Asli et al. found in 2017 a $_3F_2$ hypergeometric function representation. In this paper it is shown that the "new" orthogonal polynomials belong to the class of the Hahn polynomials. The transfer function of the unbiased FIR filter has been determined. In the second part, an orthogonal unbiased FIR filter is d…
We characterize all the multiple orthogonal threefold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as $2$-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterizati…
Explicit expressions for the Hahn multiple polynomials of type I, in terms of Kampé de Fériet hypergeometric series, are given. Orthogonal and biorthogonal relations are proven. Then, part of the Askey scheme for multiple orthogonal polynomials type I is completed. In particular, explicit expressions in terms of generalized hypergeometric series and Kampé de Fériet hypergeometric series, are given for the multiple orthogonal polynomials of type I for the Jacobi-Piñeiro, Meixner I, Meixner II, Kravchuk, Laguerre I, Laguerre II and Charlier families.
Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in $\RR^d$. Their structures are studied through a detailed analysis of classical Hahn polynomials with negative integer parameters. Factorization of the Hahn polynomials is explored and used to explain the relation between the index set of orthogonal polynomials and the lattice set in polyhedral domain. In the multivariate case, these constructions lead to nontrivial families of hypergeometric polynomials vanishing on lattice polyhedra. The generating functions and bispectral …
Let ${\bf u}$ be a nonzero linear functional acting on the space of polynomials. Let $\mathbf{D}_{q,ω}$ be a Hahn operator acting on the dual space of polynomials. Suppose that there exist polynomials $ϕ$ and $ψ$, with $\mathrm{deg}\,ϕ\leq2$ and $\mathrm{deg}\,ψ\leq1$, so that the functional equation $$ \mathbf{D}_{q,ω}(ϕ{\bf u})=ψ{\bf u} $$ holds, where the involved operations are defined in a distributional sense. In this note we state necessary and sufficient conditions, involving only the coefficients of $ϕ$ and $ψ$, such that ${\bf u}$ is regular, that is, there exists a sequence of ortho…
The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, …
We investigate the interlacing of zeros of polynomials of different degrees within the sequences of $q$-Laguerre polynomials $\left\{\tilde{L}_n^{(δ)}(z;q)\right\}_{n=0}^{\infty}$ characterized by $δ\in(-2,-1).$ The interlacing of zeros of quasi-orthogonal polynomials $\tilde{L}_n^{(δ)}(z;q)$ with those of the orthogonal polynomials $\tilde{L}_m^{(δ+t)}(z;q), m,n\in\mathbb{N}, t\in\{1,2\}$ is also considered. New bounds for the least zero of the (order $1$) quasi-orthogonal $q$-Laguerre polynomials are derived.
We construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second order difference or differential operator. The most apparent difference between classical or classical discrete orthogonal polynomials and their exceptional counterparts is that the exceptional families have gaps in their degrees, in the sense that not all degrees are present in the sequence of polynomials. The new examples have the novelty that they depend on an arbitrary number of continuous parameters.
We study a limiting case of the Askey-Wilson polynomials when one of the parameters goes to infinity, namely continuous dual q-Hahn polynomials when q > 1. Solutions to the associated indeterminate moment problem by general theory are found and an orthogonality relation is established.
We construct new examples of bispectral dual Hahn polynomials, i.e., orthogonal polynomials with respect to certain superposition of Christoffel and Geronimus transforms of the dual Hahn measure and which are also eigenfunctions of a higher order difference operator. The new examples have the novelty that they depend on an arbitrary number of continuous parameters. These are the first examples with this property constructed from the classical discrete families.
In this contribution we deal with sequences of polynomials orthogonal with respect to a Sobolev type inner product. A banded symmetric operator is associated with such a sequence of polynomials according to the higher order difference equation they satisfy. Taking into account the Darboux transformation of the corresponding matrix we deduce the connection with a sequence of orthogonal polynomials associated with a Christoffel perturbation of the measure involved in the standard part of the Sobolev inner product. A connection with matrix orthogonal polynomials is stated. The Laguerre-Sobolev ty…
We investigate the type I and type II multiple orthogonal polynomials on an $r$-star with weight function $|x|^βe^{-x^r}$, with $β>-1$. Each measure $μ_j$, for $1\leq j \leq r$, is supported on the semi-infinite interval $[0,ω^{j-1}\infty)$ with $ω=e^{2πi/r}$. For both the type I and the type II polynomials we give explicit expressions, the coefficients in the recurrence relation, the differential equation and we obtain the asymptotic zero distribution of the polynomials on the diagonal. Also, we give the connection between the Laguerre-Angelesco polynomials and the Jacobi-Angelesco polynomial…
We study algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) with respect to a weight matrix of the form $W^{(ν)}_ϕ(x) = x^νe^{-ϕ(x)} W^{(ν)}_{pol}(x)$, where $ν>0$, $W^{(ν)}_{pol}(x)$ is certain matrix valued polynomial and $ϕ$ an entire function. We introduce a pair differential operators $\mathcal{D}$, $\mathcal{D}^{\dagger}$ which are mutually adjoint with respect to the matrix inner product induced by $W^{(ν)}_ϕ(x)$. We prove that the Lie algebra generated by $\mathcal{D}$ and $\mathcal{D}^{\dagger}$ is finite dimensional if and only if…
A Survey on q-Polynomials and their Orthogonality Properties
本文研究了位于q-Hahn表（大q-雅可比多项式（bqJ））和Nikiforov-Uvarov表（Askey-Wilson多项式（AW））顶部的经典q-正交多项式对几乎任何复参数值和所有非负整数次数所满足的正交性条件.我们状态的退化版本的Favard定理，这是一个关键的文件，使我们能够扩展的正交性有效的一些整数度N的Sobolev型正交属性。我们还提出了，在[16]中应用的类似过程之后，具有满足有限正交性质的那些族的分解和离散Sobolev型正交性质的表，即它包括有限数量的质量点的总和，例如q-Racah（qR），q-Hahn（qH），对偶q-Hahn（dqH）和q-Krawtchouk多项式（qK）等。-- [16] R. S. Costas-Santos和J.F.桑切斯-拉拉。离散经典正交多项式在正交性之外的推广。J. Comp. Appl. Math.，225（2）（2009），440-451
We study the Fourier orthogonal expansions with respect to the Laguerre type weigh functions on the conic surface of revolution and the domain bounded by such a surface. The main results include a closed form formula for the reproducing kernels, which is the kernel of the orthogonal projection operator and a pseudo convolution structure on the conic domain; the latter is shown to be bounded in an appropriate $L^p$ space and used to study mean convergence of the Cesàro means of the Laguerre expansions on conic domains.
This paper presents a first result of a long term research project dealing with the construction of d-orthogonal polynomials with Hahn's property. We shall show that the latter class could be characterized by expanding a polynomial as a finite sum of first derivatives of the elements of the sequence and we shall explain how this characterization could be used to construct Hahn-classical d-orthogonal polynomials as well. In this paper we look for solutions of linear combinations of the first derivatives of two consecutive elements of the sequence by considering the derivative operator and Delta…
The problem of finding measures whose orthogonal polynomials are also eigenfunctions of higher-order difference operators have been recently solved by multiplying the classical discrete measures by suitable polynomials. This problem was raised by Richard Askey in 1991. The case of the Hahn family is specially rich due to the number of parameters involved. These suitable polynomials are in the Hahn case generated from a quartet $F_1,F_2,F_3$ and $F_4$ of finite sets of positive integers. In this paper, we show that the order of the corresponding difference operators can be reduced by assuming c…
