Nico M. Temme, Jose L. Lopez
Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the asymptotic representation of the Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Laguerre polynomials.
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In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey-Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. From the terminating basic hypergeometric representations of these polynomials, and due to symmetry in their free parameters, we are able to exhaustively explore the terminating basic hypergeometric transformation formulae which these polynomials satisfy.
We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relat…
We use connection relations and series rearrangement to generalize generating functions for several higher continuous orthogonal polynomials in the Askey scheme, namely the Wilson, continuous dual Hahn, continuous Hahn, and Meixner-Pollaczek polynomials. We also determine corresponding definite integrals using the orthogonality relations for these polynomials.
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 "Diophantine" property of the zeros of certain polynomials in the Askey scheme, recently discovered by Calogero and his collaborators, is explained, with suitably chosen parameter values, in terms of the summation theorem of hypergeometric series. Here the Diophantine property refers to integer valued zeros. It turns out that the same procedure can also be applied to polynomials arising from the basic hypergeometric series. We found, with suitably chosen parameters and certain $q-$analogue of the summation theorems, zeros of these polynomials explicitly, which are no longer integer valued.…
We produce two-dimensional contiguous relations for generalized hypergeometric functions by starting with linearization coefficients for some continuous generalized hypergeometric orthogonal polynomials in the Askey-scheme.
We present a rational extension of Newton diagram for the positivity of ${}_1F_2$ generalized hypergeometric functions. As an application, we give upper and lower bounds for the transcendental roots $β(α)$ of \begin{align*} \int_0^{j_{α, 2}} t^{-β} J_α(t) dt = 0\qquad(-1<α\le 1/2), \end{align*} where $j_{α, 2}$ denotes the second positive zero of Bessel function $J_α$.
Product formulas for basic hypergeometric series by evaluations of Askey--Wilson polynomials
本文通过阿斯凯-威尔逊多项式评估，推导出基本超几何级数的乘积公式，提出新的终止平衡${}_4ϕ_3$求和公式，并给出新的基本超几何变换和积分表示。
In this paper, we explore the symmetric nature of the terminating basic hypergeometric series representations of the Askey--Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. In particular we identify and classify the set of 4 and 7 equivalence classes of terminating balanced ${}_4ϕ_3$ and terminating very-well poised ${}_8W_7$ basic hypergeometric series which are connected with the Askey--Wilson polynomials. We study the inversion properties of these equivalence classes and also identify the connection of both sets of equ…
We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integ…
We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newton's interpolation formula.
We construct a set $H$ of orthogonal polynomial sequences that contains all the families in the Askey scheme and the $q$-Askey scheme. The polynomial sequences in $H$ are solutions of a generalized first-order difference equation which is determined by three linearly recurrent sequences of numbers. Two of these sequences are solutions of a difference equation of order three and the other sequence satisfies a related difference equation of order five. We obtain explicit expressions for the coefficients of the orthogonal polynomials and for the generalized moments with respect to a basis of Newt…
In this paper the general Krall-type Askey-Wilson polynomials are introduced. These polynomials are obtained from the Askey-Wilson polynomials via the addition of two mass points to the weight function of them at the points $\pm1$. Several properties of such new family are considered, in particular the three-term recurrence relation and the representation as basic hypergeometric series.
Terminating representations, transformations and summations for the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials
本文详尽探讨了Askey--Wilson多项式的$q$和$q^{-1}$对称子族的终止基本超几何表示与变换公式，并据此研究了其终止变换结构及$q$-Askey方案的对称群结构。
The purpose of this article is to bring structure to (basic) hypergeometric biorthogonal systems, in particular to the q-Askey scheme of basic hypergeometric orthogonal polynomials. We aim to achieve this by looking at the limits as p->0 of the elliptic hypergeometric biorthogonal functions from Spiridonov, with parameters which depend in varying ways on p. As a result we get 38 systems of biorthogonal functions with for each system at least one explicit measure for the bilinear form. Amongst these we indeed recover the q-Askey scheme. Each system consists of (basic hypergeometric) rational fu…
Askey-Wilson continued fraction at $q^N=1$
将$q^N=1$时Askey-Wilson多项式对应的连分数显式表示为两个终止$q$-超几何级数的比值。
On Askey's extension of Clausen's identity and its polynomial perturbation
本文扩展了Clausen恒等式，通过多项式扰动将超几何级数的平方转换为乘积形式，并给出了显式表达式。
The study of $-1$ orthogonal polynomials viewed as $q =-1$ limits of the $q$-orthogonal polynomials is pursued. This paper present the continuous polynomials part of the $-1$ analog of the $q$-Askey scheme. A compendium of the properties of all the continuous $-1$ hypergeometric polynomials and their connections is provided.
The Askey-Wilson polynomials are orthogonal polynomials in $x = \cos θ$, which are given as a terminating $_4ϕ_3$ basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in $z=e^{iθ}$, which are given as a sum of two terminating $_4ϕ_3$'s. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single $_4ϕ_3$'s which are Laurent polynomials in $z$ are given, which imply the non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study…
We present a method to obtain weight functions associated with linear and quadratic lattices that yield discrete orthogonality with respect to a quasi-definite moment functional of the orthogonal polynomial sequences in the Askey scheme, with the exception of the Jacobi, Bessel, Laguerre, and Hermite polynomials.
Classical discrete multiple orthogonal polynomials: hypergeometric and integral representations
本文研究任意权重的经典离散多重正交多项式，推导了Hahn多项式递归系数的显式表达式，利用多重Askey方案为各后裔族提供了I型多项式及递归系数的超几何与积分表示，并补充了相应的极限以完善该方案。
In this article we extend the results of our article "Limits of elliptic hypergeometric biorthogonal functions" to the multivariate setting. In that article we determined which families of biorthogonal functions arise as limits from the elliptic hypergeometric biorthogonal functions from Spiridonov when p->0. Here we show that the classification of the possible limits of the BC_n type multivariate biorthogonal functions previously introduced by the second author is identical to the univariate classification. That is, for each univariate limit family there exists a multivariate extension, and i…
An extended scheme of classical special functions
本文提出一个统一的超几何型经典特殊函数扩展方案，扩展了 Askey 方案及其 q-类似，顶层包含由椭圆超几何级数构成的双正交函数，并将复超几何函数纳入其中。
We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one free parameter to them. In particular, we generalize generating functions for the Askey-Wilson, continuous $q$-ultraspherical/Rogers, little $q$-Laguerre/Wall, and $q$-Laguerre polynomials. Depending on what type of orthogonality these polynomials satisfy, we derive corresponding definite integrals, infinite series, bilateral infinite series, and $q$-integrals.
Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q$-Askey scheme together with their $q$-hypergeometric representations by three sequences $x_k, h_k, g_k$ of Laurent polynomials in $q^k$, two of degree 1 and one of degree 2, satisfying certain constraints. This gives rise to a precise classification and parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also describe the four-manifold structure underlying the scheme.
A complete characterization is provided of Hankel matrices commuting with Jacobi matrices which correspond to hypergeometric orthogonal polynomials from the Askey scheme. It follows, as the main result of the paper, that the generalized Hilbert matrix is the only prominent infinite-rank Hankel matrix which, if regarded as an operator on $\ell^{2}(\mathbb{N}_{0})$, is diagonalizable by application of the commutator method with Jacobi matrices from the mentioned families.
The $q$ and $q^{-1}$-symmetric orthogonal polynomials in the $q$-Askey scheme, their dual polynomials and functions, orthogonality, generating functions and relations and nonterminating $q$-Chaundy double product representations
本文利用 q-Chaundy 定理研究 q-Askey 框架中 q 与 q^{-1}-对称正交多项式及其对偶，推导生成函数、终止超几何表示、二次变换及新的求和与积分公式。
In this paper, we propose a general method to express explicitly the inversion and the connection coefficients between two basic hypergeometric polynomial sets. As application, we consider some $d$-orthogonal basic hypergeometric polynomials and we derive expansion formulae corresponding to all the families within the $q$-Askey scheme.
In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is studied, and in particular we study solutions given in terms of power series at $0$ and at $\infty$. Factorisations of the corresponding operator are considered in terms of a lowering operator, which is the $q$-derivative, and the related raising operator. Next we consider the $q$-hypergeometric operator in a special case, and we show that there is a natural…
In this paper we present an addition to Askey's scheme of q-hypergeometric orthogonal polynomials involving classes of q-special functions which do not consist of polynomials only. The special functions are q-analogues of the Jacobi and Bessel function, and are Askey-Wilson functions, big q-Jacobi functions and little q-Jacobi functions and the corrresponding q-Bessel functions. The generalised orthogonality relations and the second order q-difference equations for these families are given. Limit transitions between these families are discussed. The quantum group theoretic interpretations are …
We introduce the notion of "hypergeometric" polynomials with respect to Newtonian bases. These polynomials are eigenfunctions ($L P_n(x) = λ_n P_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis $φ_n(x)$: $L φ_n(x) = λ_n φ_n(x) + τ_n(x) φ_{n-1}(x)$ with some coefficients $λ_n$, $τ_n$. We find the necessary and sufficient conditions for the polynomials $P_n(x)$ to be orthogonal. For the special cases where the sets $λ_n$ correspond to the classical grids, we find the complete solution to these conditions and observe that it leads to the most general Askey-Wilson po…
Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey-Wilson, Wilson and continuous Hahn families.
Eigenfunctions of the Askey-Wilson second order $q$-difference operator for $0<q<1$ and $|q|=1$ are constructed as formal matrix coefficients of the principal series representation of the quantized universal enveloping algebra $U_q(sl(2,\mathbb{C}))$. The eigenfunctions are in integral form and may be viewed as analogues of Euler's integral representation for Gauss' hypergeometric series. We show that for $0<q<1$ the resulting eigenfunction can be rewritten as a very-well-poised ${}_8ϕ_7$-series, and reduces for special parameter values to a natural elliptic analogue of the cosine kernel.
We construct a non-polynomial generalization of the $q$-Askey scheme. Whereas the elements of the $q$-Askey scheme are given by $q$-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars' hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev's quantum dilogarithm, Woronowicz's quantum exponential, or Kurokawa's double sine function. We present the basic properties of all the elements of the scheme, including their integral representations, joint eigenfunction properties…
We study Askey-Wilson type polynomials using representation theory of the double affine Hecke algebra. In particular, we prove bi-orthogonality relations for non-symmetric and anti-symmetric Askey-Wilson polynomials with respect to a complex measure. We give duality properties of the non-symmetric Askey-Wilson polynomials, and we show how the non-symmetric Askey-Wilson polynomials can be created from Sahi's intertwiners. The diagonal terms associated to the bi-orthogonality relations (which replace the notion of quadratic norm evaluations for orthogonal polynomials) are expressed in terms of r…
New bispectral orthogonal polynomials are obtained from an unconventional truncation of the Askey-Wilson polynomials. In the limit $q \to 1$, they reduce to the para-Racah polynomials which are orthogonal with respect to a quadratic bi-lattice. The three term recurrence relation and q-difference equation are obtained through limits of those of the Askey-Wilson polynomials. An explicit expression in terms of hypergeometric series and the orthogonality relation are provided. A $q$-generalization of the para-Krawtchouk polynomials is obtained as a special case. Connections with the $q$-Racah and …
A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidate…
Charting the $q$-Askey scheme. III. Verde-Star scheme for $q=1$
本文借助 Verde-Star 的三个多项式序列，对 $q=1$ Askey 方案中的正交多项式族、超几何表示及极限关系进行统一分类与图形化参数化。
Another Type of Forward and Backward Shift Relations for Orthogonal Polynomials in the Askey Scheme
本研究针对Askey方案中超几何正交多项式的移位关系，基于不同于传统的算子因式分解方式，推导出仅作用于参数的另一类前向与后向移位关系，丰富了正交多项式的性质研究。
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.
An Askey-Type Confluence Scheme for Hahn-Like Multiple Orthogonality
该研究将 Wolfs 的 Jacobi 型和 Laguerre 型多重正交系统嵌入统一的 Askey 型合流方案，通过 Bernstein 变换构造 Hahn 型祖先，并导出多种正交多项式族及其合流关系。
Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the norms of the polynomials in terms of the norm of the constant polynomial.
We introduce the power collection method for easily deriving connection relations for certain hypergeometric orthogonal polynomials in the $(q-)$Askey scheme. We summarize the full-extent to which the power collection method may be used. As an example, we use the power collection method to derive connection and connection-type relations for Meixner and Krawtchouk polynomials. These relations are then used to derive generalizations of generating functions for these orthogonal polynomials. The coefficients of these generalized generating functions are in general, given in term of multiple hyperg…
Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= λ_n x^n + ν_n x^{n-2}$ for $λ_n$ and $ν_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = λ_n P_n(x)$. A classification of all the cases for which the polynomials $P_n(x)$ are orthogonal is provided. A general derivation of the algebras explaining the bispectrality of the polynomials is given. The resulting algebras prove to be central extensions of the Askey-Wilson algebra and its degenerate cases.
Classification of the hypergeometric orthogonal polynomials via their recurrence coefficients
利用三项递推关系中的系数α_n和β_n对超几何正交多项式序列进行分类，得到53个不相交的类，与Askey方案不同。
Charting the $q$-Askey scheme. II. The $q$-Zhedanov scheme
本文通过定义 $q$-Zhedanov 代数并利用 Verde-Star 参数化，系统绘制了 $q$-Askey 方案中族的子族与极限族关系，并与 $q$-Verde-Star 方案对比，揭示差异。
An Askey Scheme for Jacobi Systems of Mixed Type: Laguerre Limits of the First and Second Kinds and Hermite Limits
本文为混合型Jacobi多重正交系统建立Askey汇合方案，通过变量重标度导出两类Laguerre及Hermite极限系统，给出显式极限权重、正交形式与矩收敛，并证明三条路径得到一致结果。
q-Special functions, an overview
本文概述q-特殊函数理论，涵盖q-超几何级数、Askey–Wilson多项式、Macdonald多项式及椭圆超几何级数，并讨论其在数论、组合、量子群和统计力学中的应用。
We construct 3 finite systems of $4-F-3$ hypergeometric orthogonal polynomials. The weights are 1) the weight defined by the $5-H-5$ Dougall summation formula; 2) the integrand in the Askey beta-integral; 3) the weight $w(s)=|p(s)/q(s)|^2$, where $p(s)=Γ(a+is)Γ(b+is)Γ(c+is)$ and $q(s)=Γ(d+is)Γ(2is)$. We also evaluate the integral of the function $w(s)$ using the Jacobi transform (the Olevsky transform); this integral also can be reduced to the Nassrallah--Rahman integral.
The irreducible $*$-representations of the Lie algebra $su(1,1)$ consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch-Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application…
Elements of the q-Askey scheme in the algebra of symmetric functions
本研究在对称函数代数中构造多变量q-Racah对称函数，推导其可退化为big q-Jacobi等三类对称函数，证明四类函数均关于无穷点构形空间上的测度正交，且测度间的极限转移与函数退化一致，丰富了q-Askey概型的多变量对称函数理论。
"Diophantine'' and Factorisation Properties of Finite Orthogonal Polynomials in the Askey Scheme
该研究探索Askey系中十二类有限正交多项式的丢番图与因式分解性质的新内涵，指出高次首一多项式是对应三对角矩阵的零范数特征向量，并以此为种子解构造新型多指标正交多项式，验证其形状不变性并给出显式变换公式。
This paper is to evaluate certain Catalan-Hankel Pfaffians by the theory of skew orthogonal polynomials. Due to different kinds of hypergeometric orthogonal polynomials underlying the Askey scheme, we explicitly construct the classical skew orthogonal polynomials and then give different examples of Catalan-Hankel Pfaffians with continuous and $q$-moment sequences.
We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a new type of elliptic hypergeometric function. We show that obtained polynomials are orthogonal on the unit circle with respect to a dense point meausure, i.e. the spectrum consists from infinite number points of increase which are dense on the unit circle. We construct also corresponding explicit systems of polynomials orthogonal on the interval of the real ax…
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…
On the classification of hypergeometric families of orthogonal polynomials on the real line
本文提出了一种新的方法，对实线上超几何正交多项式族进行分类，发现存在10种类型，其中8种来自Askey方案，2种为新类型的Lommel多项式相关族。
We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s \in \mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden et al. [F. D. Cunden, F. Mezzadri, N. J. S…
A new multiple-integral representation of a general family of very-well-poised hypergeometric series is proved. Inspite of an analytic character of the result, it is motivated by the recent arithmetic progress for the values of the Riemann zeta function at odd integers. The multiple integrals are of Euler-type and can be viewed as a natural generalization of Beukers' integrals for $ζ(2)$ and $ζ(3)$.
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…
For a hypergeometric series $\sum_k f(k,a, b, ...,c)$ with parameters $a, b, >...,c$, Paule has found a variation of Zeilberger's algorithm to establish recurrence relations involving shifts on the parameters. We consider a more general problem concerning several similar hypergeometric terms $f_1(k, a, b,..., c)$, $f_2(k, a,b, ..., c)$, $...$, $f_m(k, a, b, ..., c)$. We present an algorithm to derive a linear relation among the sums $\sum_k f_i(k,a,b,...,c)$ $(1\leq i \leq m)$. Furthermore, when the summand $f_i$ contains the parameter $x$, we can require that the coefficients be $x$-free. Suc…
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