This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Jacobi polynomials which are orthogonal in a weighted Hilbert function space on the the interval (-1,+1) of the real line. These polynomials are generated by a second-order ordinary linear differential equation with a spectral parameter. Some additional information on the Sturm-Liouville form of this equation is given in this note, together with details of the singular differential operators generated in the weighted Hilbert function space. In particular, structured boundary conditions are given to determine the special self-adjoint operator, whose discrete spectrum and associated eigenvectors yield the X_(1)-Jacobi polynomials.
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…
The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appell's function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are expressed in terms of generalized hypergeometric function. Some of the interesting special cases of the main results also established.
Persymmetric Jacobi matrices are invariant under reflection with respect to the anti-diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the restrictions either to the odd or to the even points of the complete orthogonality lattice. This is exploited to design very efficient inverse problem algorithms for the reconstruction of persymmetric Jacobi matrices from spectral points. Isospectral deformations of such matrices are also considered. Expressions for the associated polynomials and their weight…
Estimates for Jacobi-Sobolev type orthogonal polynomials
设Sobolev型内积<f，g> = \int fg d mu_0+ int f' g' d mu_1，其中mu_0 = w + M delta_c，mu_1= N delta_c，其中w是Jacobi权，c是1或-1，M，N >= 0。我们得到了关于<...>的正交多项式在[-1，1]上的估计和渐近性质.和它们的内核。我们还比较了这些多项式与Jacobi正交多项式。
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.…
We investigate type I multiple orthogonal polynomials on $r$ intervals which have a common point at the origin and endpoints at the $r$ roots of unity $ω^j$, $j=0,1,\ldots,r-1$, with $ω= \exp(2πi/r)$. We use the weight function $|x|^β(1-x^r)^α$, with $α,β>-1$ for the multiple orthogonality relations. We give explicit formulas for the type I multiple orthogonal polynomials, the coefficients in the recurrence relation, the differential equation, and we obtain the asymptotic distribution of the zeros.
We study a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue equation involving a third order differential operator of Dunkl-type. The orthogonality measure of these polynomials consists in the continuous measure of the little -1 Jacobi polynomials to which is added an arbitrary mass located at the point $x=0$, the middle of the orthogonality interval. This provides the first nontrivial example of Krall-type polynomials with a point mass inside the orthogonality interval. These polynomials can be obtained by a Geronimus transform of the litt…
Using Casorati determinants of Hahn polynomials $(h_n^{α,β,N})_n$, we construct for each pair $\F=(F_1,F_2)$ of finite sets of positive integers polynomials $h_n^{α,β,N;\F}$, $n\in σ_\F$, which are eigenfunctions of a second order difference operator, where $σ_\F$ is certain set of nonnegative integers, $σ_\F \varsubsetneq \NN$. When $N\in \NN$ and $α$, $β$, $N$ and $\F$ satisfy a suitable admissibility condition, we prove that the polynomials $h_n^{α,β,N;\F}$ are also orthogonal and complete with respect to a positive measure (exceptional Hahn polynomials). By passing to the limit, we transfo…
Mehler-Heine asymptotics describe the behavior of orthogonal polynomials near the edges of the interval where the orthogonality measure is supported. For Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the hard edge involves Bessel functions $J_α$. We show that the asymptotic behavior near the endpoint of the interval of (one of) the measures for multiple orthogonal polynomials involves a generalization of the Bessel function. The multiple orthogonal polynomials considered are Jacobi-Angelesco polynomials, Jacobi-Piñeiro polynomials, multiple Laguerre polynomials, mul…
We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by $α>-1$, $-2<β<-1$. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials $P_n^{(α, β)}$ and $P_{n}^{(α,β+2)}$ are interlacing, holds when the parameters $α$ and $β$ are in the range $α>-1$ and $-2<β<-1$. We prove that the zeros of $P_n^{(α, β)}$ and $P_{n+1}^{(α,β)}$ do not interlace for any $n\in\mathbb{N}$, $n\geq2$ and any fixed $α$, $β$ with $α>-1$, $-2<β<-1$. The interlacing of zeros of $P_n^{(α,β)}$ an…
We recast Byerly's formula for integrals of products of Legendre polynomials. Then we adopt the idea to the case of Jacobi polynomials. After that, we use the formula to derive an asymptotic formula for integrals of products of Jacobi polynomials. The asymptotic formula is similar to an analogous one recently obtained by the first author and Jeff Geronimo for a different case. Thus, it suggests that such an asymptotic behavior is rather generic for integrals of products of orthogonal polynomials.
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.
We study the monotonic behaviour of the zeros of the multiple Jacobi-Angelesco orthogonal polynomials, in the diagonal case, with respect to the parameters $α,β$ and $γ$. We prove that the zeros are monotonic functions of $α$ and $γ$ and consider some special cases of how the zeros depend on $β$, especially in the presence of symmetry. As a consequence we obtain results about monotonicity of zeros of Jacobi-Laguerre and Laguerre-Hermite multiple orthogonal polynomials too.
We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for large classes of orthogonality measures.
We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval $[-1,1]$ with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order equations was introduced by J. and R. Koekoek in 1999 essentially by using special function methods. In this paper, a completely elementary representation of the Jacobi-type differential operator of any even order is given. This enables us to trace the orthogonality relation of the Jacobi-type polynomials back to their differential equation. Moreover, we establ…
We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-Abelian extensions of a family of discrete Painlevé d-PIV equations are obtained for the three term recurrence relation coefficients.
For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by Koornwinder's generalized Jacobi polynomials with four parameters $α,β\in\mathbb{N}_{0}$ and $M,N \ge 0$ determining the orthogonality measure on the interval $-1 \le x \le 1$. The corresponding differential equation of order $2α+2β+6$ is presented here as a linear combination of four elementary components which make the corresponding differential operator widely ac…
We provide the mathematical foundation for the $X_m$-Jacobi spectral theory. Namely, we present a self-adjoint operator associated to the differential expression with the exceptional $X_m$-Jacobi orthogonal polynomials as eigenfunctions. This proves that those polynomials are indeed eigenfunctions of the self-adjoint operator (rather than just formal eigenfunctions). Further, we prove the completeness of the exceptional $X_m$-Jacobi orthogonal polynomials (of degrees $m, m+1, m+2, ...$) in the Lebesgue--Hilbert space with the appropriate weight. In particular, the self-adjoint operator has no …
In this note we revive a transformation that was introduced by H. S. Wall and that establishes a one-to-one correspondence between continued fraction representations of Schur, Carathéodory, and Nevanlinna functions. This transformation can be considered as an analog of the Szegő mapping but it is based on the Cayley transform, which relates the upper half-plane to the unit disc. For example, it will be shown that, when applying the Wall transformation, instead of OPRL, we get a sequence of orthogonal rational functions that satisfy three-term recurrence relation of the form $(H-λJ)u=0$, where …
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.
Computing asymptotics of the recurrence coefficients of X1-Jacobi polynomials we investigate the limit of Christoffel function. We also study the relation between the normalized counting measure based on the zeros of the modified average characteristic polynomial and the Christoffel function in limit. The proofs of corresponding theorems with respect to ordinary orthogonal polynomials are based on the three-term recurrence relation. The main point is that exceptional orthogonal polynomials possess at least five-term formulae and so the Christoffel-Darboux formula also fails. It seems that thes…
We prove that similarly to the standard case, the equilibrium measure of Julia sets of exceptional Jacobi polynomials tends to the equilibrium measure of the interval of orthogonality in weak-star sense.
In this paper, we consider a natural extension of several results related to Krall-type polynomials introducing a modification of a $q$-classical linear functional via the addition of one or two mass points. The limit relations between the $q$-Krall type modification of big $q$-Jacobi, little $q$-Jacobi, big $q$-Laguerre, and other families of the $q$-Hahn tableau are established.
In this paper we study a generalization of the class of orthogonal polynomials on the real line. These polynomials satisfy the following relation: $(J_5 - λJ_3) \vec p(λ) = 0$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal, $\vec p(λ) = (p_0(λ), p_1(λ), p_2(λ),\cdots)^T$, the superscript $T$ means the transposition, with the initial conditions $p_0(λ) = 1$, $p_1(λ) = αλ+ β$, $α> 0$, $β\in\mathbb{R}$. Some orthonormality conditions for the polynomials $\{ p_n(λ) \}_{n=0}^\infty$ are obtained. An ex…
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.
The purpose of this note is to extend in a simple and unified way some results on orthogonal polynomials with respect to the weight function $$\frac{|T_m(x)|^p}{\sqrt{1-x^2}}\;,\quad-1<x<1\;,$$ where $T_m$ is the Chebyshev polynomial of the first kind of degree $m$ and $p>-1$.
We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green's functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on $\mathbb{Z}_+$ to higher dimension. We illustrate importance of this connection by proving…
We provide an explicit spectral representation for several weighted Hankel matrices by means of the so called commutator method. These weighted Hankel matrices are found in the commutant of Jacobi matrices associated with orthogonal polynomials from the Askey scheme whose Jacobi parameters are polynomial functions of the index. We also present two more results of general interest. First, we give a complete description of the commutant of a Jacobi matrix. Second, we deduce a necessary and sufficient condition for a weighted Hankel matrix commuting with a Jacobi matrix to determine a unique self…
