Howard S. Cohl, Roberto S. Costas-Santos
We study special values for the continuous $q$-Jacobi polynomials and present applications of these special values which arise from bilinear generating functions, and in particular the Poisson kernel for these polynomials.
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We study the bispectrality of Jacobi type polynomials, which are eigenfunctions of higher-order differential operators and can be defined by taking suitable linear combinations of a fixed number of consecutive Jacobi polynomials. Jacobi type polynomials include, as particular cases, the Krall-Jacobi polynomials. As the main results we prove that the Jacobi type polynomials always satisfy higher-order recurrence relations (i.e., they are bispectral). We also prove that the Krall-Jacobi families are the only Jacobi type polynomials which are orthogonal with respect to a measure on the real line.
Some weighted inequalities for the maximal operator with respect to the discrete diffusion semigroups associated with exceptional Jacobi and Dunkl-Jacobi polynomials are given. This setup allows to extend the corresponding results obtained for discrete heat semigroup recently to richer class of differential-difference operators.
A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials. Also the limits from Askey-Wilson to Wilson polynomials and from q-Racah to Racah polynomials are given in a more conceptual way.
Multiple Wilson and Jacobi-Pineiro polynomials
我们引入了多重Wilson多项式，给出了II型多重正交多项式（Hermite-Pade多项式）的一个新例子。这些多项式可以被写为Jacobi-Pineiro变换，其是由T.H.库恩温德在这里，我们需要介绍Jacobi和Jacobi-Pineiro多项式与复杂的参数。给出了Jacobi-Pineiro多项式和多重Wilson多项式的显式公式，其中一个是Kampe de Feriet级数。最后，我们看看一些限制关系，并构建一个多AT-askey表的一部分。
A class of matrix-valued polynomials generalizing Jacobi Polynomials
得到了一类推广雅可比多项式的矩阵值多项式族，它们由Rodrigues公式定义，也是一列微分算子的乘积。每一类多项式都是完备的，满足两步递推关系、积分相关关系和拟正交关系。
Upward extension of the Jacobi matrix for orthogonal polynomials
实直线上的正交多项式总是满足三项递推关系。递归系数确定了唯一表征正交多项式的三对角半无限矩阵（雅可比矩阵）。我们研究新的正交多项式通过添加到雅可比矩阵$r$新的行和列，使原来的雅可比矩阵向下移动。$r$新的行和列包含$2r$新的参数，并且新获得的正交多项式因此对应于雅可比矩阵的向上扩展。利用原正交多项式及其相关多项式和$2r$新参数，给出了新的正交多项式的显式表达式，并在原正交多项式为经典多项式时，给出了新多项式的四阶微分方程.此外，我们展示了如何正交化措施，这些新的正交多项式可以获得和工作的细节，为一个单参数家庭的雅可比多项式的相关多项式再次雅可比多项式。
Is is shown in this paper that there is a connection between the Riemann zeta-function $\zf$ and the classical Jacobi's polynomials, i.e. the Legendre polynomials, Chebyshev polynomials of the first and the second kind,...
On Jacobi and continuous Hahn polynomials
利用傅里叶变换将Jacobi多项式映射到连续Hahn多项式上，由雅可比多项式的正交关系和Parseval公式导出连续Hahn多项式的正交关系.在一个特殊的情况下，这种关系可以追溯到工作贝特曼在1933年，我们遵循的一部分历史发展，这些多项式。给出了该关系的一些应用。
This is a tutorial introduction to the representation theory of SU(2) with emphasis on the occurrence of Jacobi polynomials in the matrix elements of the irreducible representations. The last section traces the history of the insight that Jacobi polynomials occur in the representation theory of SU(2).
We study a new family of "classical" orthogonal polynomials, here called big -1 Jacobi polynomials, which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with differential operators of Dunkl-type. These polynomials can be obtained from the big $q$-Jacobi polynomials in the limit $q \to -1$. An explicit expression of these polynomials in terms of Gauss' hypergeometric functions is found. The big -1 Jacobi polynomials are orthogonal on the union of two symmetric intervals of the real axis. We show that the big -1 Jacobi polynomials can be obtained from the Bannai-Ito poly…
In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generating functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given.
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。
Multiple little q-Jacobi polynomials
通过对指数格q^k（k= 0，1，2，3，.）上的r个离散小q-Jacobi测度施加正交条件，引入了两类多重小q-雅可比多项式，其中0 < q < 1。我们发现，这些多个小q-雅可比多项式有有用的q-差分性质，如罗德里格斯公式（由r个差分算子的乘积组成）。给出了这些多项式零点的一些性质和渐近性质。
Jacobi polynomials of type BC, Jack polynomials, limit transitions and O(\infty)
这是在“对称空间上的傅立叶和拉东变换”会议上为纪念S. 2002年9月25日，丹麦罗斯基勒，赫尔加松65岁生日。1992年10月12日
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.
Asymptotics for the simplest generalized Jacobi polynomials recurrence coefficients from Freud's equations: numerical explorations.
广义雅可比多项式是与权函数相关的正交多项式，该权函数在直到有限个点的整个正交区间上是光滑的和正的，其中出现代数奇点。这些奇点的递归系数的渐近行为的影响进行了调查。
Hahn polynomials of several variables can be defined by using the Jacobi polynomials on the simplex as a generating function. Starting from this connection, a number of properties for these two families of orthogonal polynomials are derived. It is shown that the Hahn polynomials appear as connecting coefficients between several families of orthogonal polynomials on the simplex. Closed-form formulas are derived for the reproducing kernels of the Hahn polynomials and Krawtchouck polynomials. As an application, the Poisson kernels for the Hahn polynomials and the Krawtchouck polynomials are shown…
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…
Matrix-valued analogues of the little q-Jacobi polynomials are introduced and studied. For the 2x2-matrix-valued little q-Jacobi polynomials explicit expressions for the orthogonality relations, Rodrigues formula, three-term recurrence relation and their relation to matrix-valued q-hypergeometric series and the scalar-valued little q-Jacobi polynomials are presented. The study is based on a matrix-valued q-difference operator, which is a q-analogue of Tirao's matrix-valued hypergeometric differential operator.
In this paper we present a systematic way to describe exceptional Jacobi polynomials via two partitions. We give the construction of these polynomials and restate the known aspects of these polynomials in terms of their partitions. The aim is to show that the use of partitions is an elegant way to label these polynomials. Moreover, we prove asymptotic results according to the regular and exceptional zeros of these 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…
Asymptotic approximations of Jacobi polynomials are given for large values of the $β$-parameter and of their zeros. The expansions are given in terms of Laguerre polynomials and of their zeros. The levels of accuracy of the approximations are verified by numerical examples.
Jacobi Polynomials from Compatibility Conditions
我们重新讨论了正交多项式的梯形算子，并重新解释了两个补充条件作为两个线性超定系统的相容性条件;一个涉及多项式关于变量z（谱参数）的变化，另一个涉及n（格变量）中的递归关系。对于[-1，1]中Jacobi权w（x）=（1-x）^a（1+x）^b，x，我们给出了如何利用相容性条件显式地确定一元雅可比多项式的递推系数.
The CMV bispectrality of the Jacobi polynomials on the unit circle
本文证明了单位圆上正交的Jacobi多项式具有CMV双谱性，即满足递推关系和Dunkl型微分方程，并引入圆Jacobi代数作为其隐藏对称代数，以推导其所有基本性质。
Inversion methods for finding differential equations for generalized Jacobi polynomials
我们寻找广义雅可比多项式所满足的微分方程，这些多项式在区间[-1，1]上关于由经典Jacobi权函数和正交区间端点上的点质量组成的权函数正交.为了找到这些微分方程的系数的显式公式，我们必须求解包含经典雅可比多项式的导数的方程组。我们表明，这些系统的方程有一个明确给出的唯一的解决方案。这是本文证明的Jacobi反演公式的一个推论。
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.
Asymptotic approximations of Jacobi polynomials are given in terms of elementary functions for large degree $n$ and parameters $α$ and $β$. From these new results, asymptotic expansions of the zeros are derived and methods are given to obtain the coefficients in the expansions. These approximations can be used as initial values in iterative methods for computing the nodes of Gauss--Jacobi quadrature for large degree and parameters. The performance of the asymptotic approximations for computing the nodes and weights of these Gaussian quadratures is illustrated with numerical examples.
We give the asymptotic distribution of the zeros of Jacobi-Piñeiro polynomials and multiple Laguerre polynomials of the first kind. We use the nearest neighbor recurrence relations for these polynomials and a recent result on the ratio asymptotics of multiple orthogonal polynomials. We show how these asymptotic zero distributions are related to the Fuss-Catalan distribution.
Below the normalized weighted reciprocal of the Christoffel function with respect to exceptional Jacobi polynomials is investigated. It is proved that it tends to the equilibrium measure of the interval of orthogonality in weak-star sense. The main tool of this study is the multiplication operator and examination of behavior of zeros of the corresponding average characteristic polynomial. Finally, as an application of multiplication operator, location of zeros of certain self-inversive polynomials are examined.
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.
Simple bounds for the extreme zeroes of Jacobi polynomials
该研究用初等方法推导雅可比多项式极值零点的简单新界，其盖根鲍尔多项式最大零点下界可媲美现有最优结果，形式简洁便于使用。
The $X_1$-Jacobi and the $X_1$-Laguerre exceptional orthogonal polynomials have been introduced and studied by Gómez-Ullate, Kamran and Milson in a series of papers. In this note, we establish some properties, such as interlacing, monotonicity with respect to the parameters and order, about the so-called \textit{regular} and \textit{exceptional} zeros of these two classes of polynomials.
Some results on co-recursive associated Laguerre and Jacobi polynomials
我们提出的结果的共同递归相关的Laguerre和雅可比多项式，这是感兴趣的一些出生和死亡过程的Chapman-Kolmogorov方程的解决方案，或没有吸收。给出了该谱测度的显式形式、生成函数和绝对连续部分。我们导出了多项式所满足的四阶微分方程，特别注意了一些简单的极限情况。
On Polar Jacobi Polynomials
本文考察极坐标 Jacobi 多项式，重点分析其相对于基于 Jacobi 测度的离散-连续 Sobolev 型内积的代数和微分性质。
We show asymptotic expansions of the eigenfunctions of certain perturbations of the Jacobi operator in a bounded interval, deducing equiconvergence results between expansions with respect to the associated orthonormal basis and expansions with respect to the cosine basis. Several results for pointwise convergence then follow.
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.
Szmytkowski derived a certain integral with Gegenbauer polynomials. A natural generalization is to derive lookalike integrals with Jacobi polynomials. Six methods are treated to derive the first integral. The first method should be enough to prove the first integral, but by the other methods there arises remarkable formula such as par example a zero-balanced F3 Appell function which can be converted into a 2F1 hypergeometric function. Another three integrals complete the paper.
Error bounds for the asymptotic expansions of the Jacobi polynomials
本文用积分方法为Jacobi多项式次数趋于无穷时的渐近展开推导显式误差界，聚焦外部振荡区域，引入新技术处理对数奇异性，并给出系数递推公式。
Orthogonality of quasi-nature spectral polynomials of Jacobi and Laguerre type
本文确定Jacobi多项式的一阶拟Christoffel和拟Geronimus多项式系数显式表达式，建立其拟谱多项式正交性，推导Laguerre型正交性，分析零点交错性质，并扩展至单位圆。
Darboux-type formula for Jacobi biorthogonal polynomials
本文研究了Jacobi双正交多项式的渐近行为，利用最陡下降法建立Darboux型公式，并通过构造适当积分路径应用Rodrigues公式，结果在正交情况下退化为经典Darboux公式。
Hypergeometric Expressions for Type I Jacobi-Piñeiro Orthogonal Polynomials with Arbitrary Number of Weights
本文针对任意数量测度，利用多重超几何函数给出了I型Jacobi-Piñeiro和第一类Laguerre多重正交多项式的显式表达式。
We investigate asymptotic behavior of polynomials $ Q_n(z) $ satisfying non-Hermitian orthogonality relations $$ \int_Δs^kQ_n(s)ρ(s)\dd s =0, \quad k\in\{0,\ldots,n-1\}, $$ where $ Δ:= [-a,a]\cup [-\ic b,\ic b] $, $ a,b>0 $, and $ ρ(s) $ is a Jacobi-type weight.
Perturbations of Jacobi polynomials and piece-wise hypergeometric orthogonal systems
我们在射线$[0,\infty)$上构造了非完全正交系，它们在$n\mapsto n+a$阶移动后看起来像雅可比多项式$P_n(x)$，其中$a$是实常数。这些方程组是超几何微分算子奇异Sturm-Liouville问题的解。我们得到了这些问题的显式谱分解。
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…
Note on the X_(1)-Jacobi orthogonal polynomials
本文补充了Gomez-Ullate，Kamran和Milson关于实直线区间（-1，+1）上加权Hilbert函数空间中正交的X_（1）-雅可比多项式的工作。这些多项式由一个带谱参数的二阶常线性微分方程生成。本文给出了该方程的Sturm-Liouville形式的一些附加信息，以及在加权Hilbert函数空间中生成的奇异微分算子的细节。特别地，给出了确定特殊自伴算子的结构化边界条件，其离散谱和相关特征向量产生X_（1）-雅可比多项式.
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 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.
We will examine the electrostatic properties of exceptional and regular zeros of $X_m$-Laguerre and $X_m$-Jacobi polynomials. Since there is a close connection between the electrostatic properties of the zeros and the stability of interpolation on the system of zeros, we can deduce an Egerváry-Tur\' an type result as well. The limit of the energy on the regular zeros are also investigated.
A few identities and integrals involving Pochhammer symbols, Jacobi polynomials, and the generalized hypergeometric function
本文通过Jacobi多项式和广义超几何函数的展开，推导出涉及Pochhammer符号的新的恒等式和积分闭形式。
On Bivariate Jacobi Konhauser Polynomials
本文基于单变量双正交与正交多项式构造方法，提出新型双变量Jacobi Konhauser多项式，研究其基本性质与变换算子下的像，并定义对应JKML函数及相关性质。
Orthogonality of Jacobi and Laguerre polynomials for general parameters via the Hadamard finite part
对复形a，b（a，b不是负整数，a+b不是-2，-3，.）使用积分的Hadamard有限部分，在经典情况下给出它们的正交性。找到了这些多项式所满足的黎曼-希尔伯特问题，其结果与经典情形（当a，b的实部大于-1时）的结果在形式上类似
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…
By applying the Euler--Rayleigh methods to a specific representation of the Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, we derive upper and lower bound for $1-x_{nn}^2(λ)$, with $x_{nn}(λ)$ being the largest zero of the $n$-th ultraspherical polynomial $P_n^{(λ)}$. For every fixed $λ>-1/2$, the limit of the ratio of our upper and lower bounds for $1-x_{nn}^2(λ)$ does not exceed $1.6$. This paper is a continuation of [1].
Using the duplication formulas of the elliptic trigonometric functions of Gosper, we deduce some new special values for the first two Jacobi theta functions. At the end of the paper, we show how is it possible to extend our arguments and deduce a wide variety of additional special values for the Jacobi thetas. In addition, an identity is revealed between these functions and the Weber modular function.
Localized polynomial frames on the interval with Jacobi weights
众所周知，在具有$w_{\a,\b}(t) = (1-t)^\a(1+t)^\b$的$L^2(w_{\a,\b}, [-1, 1])$中的次数为$n$的多项式上的正交投影器的核可以根据雅可比多项式来写。它示出，如果在这个内核中的系数被平滑通过采样的$C^\infty$功能，然后得到的功能几乎指数（比任何多项式更快）的衰减率远离主对角线。这个结果是用于建设紧多项式框架$L^2(w_{\a,\b})$的元素几乎指数本地化。
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.…
For a two-parameter family of lower triangular matrices with entries involving Jacobi polynomials an explicit inverse is given, with entries involving a sum of two Jacobi polynomials. The formula simplifies in the Gegenbauer case and then one choice of the parameter solves an open problem in a recent paper by Koelink, van Pruijssen & Roman. The two-parameter family is closely related to two two-parameter groups of lower triangular matrices, of which we also give the explicit generators. Another family of pairs of mutually inverse lower triangular matrices with entries involving Jacobi polynomi…
Two-point completion of zero interlacing and Wronskian-type bounds, with applications to Jacobi and other orthogonal polynomials
本文研究了无法恰好交错两个点的多项式序列的完整交错，通过一般混合递推关系确定了两个额外点的位置，应用于正交多项式家族并改进了Jacobi多项式和Meixner-Pollaczek多项式的研究结果。
Convergent Asymptotic Expansions of Charlier, Laguerre and Jacobi Polynomials
推导了三种类型的正交多项式：Charlier，Laguerre和Jacobi的收敛展开式.该展开式对于大的度值具有渐近性质。展开式以函数的形式给出，这些函数是给定多项式的特例。该方法基于在复平面的一个或两个点上展开积分，这些点是被积函数的相函数的鞍点。
Differential equations for generalized Jacobi polynomials
我们寻找广义雅可比多项式满足的谱型微分方程，这些多项式关于由经典Jacobi权函数和在正交区间端点的两个点质量组成的权函数在区间[-1，1]上正交.我们证明了这样一个微分方程是唯一确定的，我们给出了明确的表示系数。在非零质量点的情况下，这个微分方程的阶是无限的，除了（其中一个）参数的非负整数值。另外，还以参数的形式给出了系统的有穷级.
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