arXiv subjects

Tom H. Koornwinder

Publications and source records attributed to Tom H. Koornwinder.

At least 19 recordsLinked to original sources

arXiv2026

Automorphisms of the DAHA of type $\check{C_1}C_1$ and non-symmetric Askey-Wilson functions

In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type $\check{C_1}C_1$ which have a relatively simple action on the generators and on the parameters, notably a symmetry $t_4$ which sends the Askey-Wilson parameters $(a,b,c,d)$ to $(a,b,qd^{-1},qc^{-1})$. We study how these symmetries act on the basic representation and on the symmetric and non-symmetric Askey-Wilson (AW) polynomials and functions. Interestingly $t_4$ maps AW polynomials to functions. We take the rank one case of Stokman's Cherednik kernel for $BC_n$ as the definition of the non-symmetric Askey--Wilson function. From it we derive an expression as a sum of a symmetric and an anti-symmetric term.

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arXiv2026

Dick and Liz Askey's visit to U.S.S.R. in 1987, and how the discrete Askey scheme also originated in Russia

This paper describes how the discrete Askey scheme independently arose in Russia and how Askey learned about this. In particular, Askey met main characters in this story, namely Gel'fand and Suslov as well as Nikiforov and Uvarov, during his trip to U.S.S.R, in September 1987. The paper describes this trip in some detail, in particular based on the diary of Askey's wife Liz, who accompanied him. Dick and Liz Askey continued their trip by visits to Japan, Australia and India. Schedules of these three visits are also given.

math.HO↗

arXiv2025

Additions to the formula lists in "Hypergeometric orthogonal polynomials and their $q$-analogues" by Koekoek, Lesky and Swarttouw

This paper gives a rather arbitrary choice of formulas for ($q$-)hypergeometric orthogonal polynomials which the author missed while consulting Chapters 9 and 14 in the book "Hypergeometric orthogonal polynomials and their $q$-analogues" by Koekoek, Lesky and Swarttouw. The systematics of these chapters will be followed here, in particular for the numbering of subsections and of references.

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arXiv2025

On an identity of Chaundy and Bullard. III. Basic and elliptic extensions

The identity by Chaundy and Bullard expresses $1$ as a sum of two truncated binomial series in one variable where the truncations depend on two different non-negative integers. We present basic and elliptic extensions of the Chaundy--Bullard identity. The most general result, the elliptic extension, involves, in addition to the nome $p$ and the base $q$, four independent complex variables. Our proof uses a suitable weighted lattice path model. We also show how three of the basic extensions can be viewed as Bézout identities. Inspired by the lattice path model, we give a new elliptic extension of the binomial theorem, taking the form of an identity for elliptic commuting variables. We further present variants of the homogeneous form of the identity for $q$-commuting and for elliptic commuting variables.

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arXiv2024

Charting the $q$-Askey scheme. III. Verde-Star scheme for $q=1$

Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q=1$ Askey scheme together with their hypergeometric representations by three sequences $x_k, h_k, g_k$ of polynomials in $k$, two of degree 2 and one of degree 4, satisfying certain constraints. Except for the Hermite polynomials, this gives rise to a precise classification and a very simple uniform parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also discuss limits from the $q$-case to the case $q=1$, although this cannot be done in a uniform way.

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arXiv2024

Josef Meixner: his life and his orthogonal polynomials

This paper starts with a biographical sketch of the life of Josef Meixner. Then his motivations to work on orthogonal polynomials and special functions are reviewed. Meixner's 1934 paper introducing the Meixner and Meixner-Pollaczek polynomials is discussed in detail. Truksa's forgotten 1931 paper, which already contains the Meixner polynomials, is mentioned. The paper ends with a survey of the reception of Meixner's 1934 paper.

math.HO↗

arXiv2023

q-Special functions, an overview

This article gives a brief introduction to $q$-special functions, i.e., $q$-analogues of the classical special functions. Here $q$ is a deformation parameter, usually $0<q<1$, where $q=1$ is the classical case. The main topics to be treated are $q$-hypergeometric series, with some selected evaluation and transformation formulas, and the $q$-hypergeometric orthogonal polynomials, most notably the Askey--Wilson polynomials. Some newer topics as nonsymmetric analogues and $q=-1$ limits will also be addressed. In several variables we discuss Macdonald polynomials associated with root systems, in particular the $A_n$ and the $BC_n$ case. The theory of elliptic hypergeometric series also gets some attention. The occurrence of $q$-series in number theory and combinatorics will be discussed. Finally we indicate applications and interpretations in quantum groups, Chevalley groups, affine Lie algebras and statistical mechanics.

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arXiv2022

Charting the $q$-Askey scheme. II. The $q$-Zhedanov scheme

This is the second in a series of papers which intend to explore conceptual ways of distinguishing between families in the $q$-Askey scheme and uniform ways of parametrizing the families. For a system of polynomials $p_n(x)$ in the $q$-Askey scheme satisfying $Lp_n=h_np_n$ with $L$ a second order $q$-difference operator the $q$-Zhedanov algebra is the algebra generated by operators $L$ and $X$ (multiplication by $x$). It has two relations in which essentially five coefficients occur. Vanishing of one or more of the coefficients corresponds to a subfamily or limit family of the Askey-Wilson polynomials. An arrow from one family to another means that in the latter family one more coefficient vanishes. This yields the $q$-Zhedanov scheme given in this paper. The $q$-hypergeometric expression of $p_n(x)$ can be interpreted as an expansion of $p_n(x)$ in terms of certain Newton polynomials. In our previous paper arXiv:2108.03858 we used Verde-Star's clean parametrization of such expansions and we obtained a $q$-Verde-Star scheme, where vanishing of one or more of these parameters corresponds to a subfamily or limit family. The actions of the operators $L$ and $X$ on the Newton polynomials can be expressed in terms of the Verde-Star parameters, and thus the coefficients for the $q$-Zhedanov algebra can be expressed in terms of these parameters. There are interesting differences between the $q$-Verde-Star scheme and the $q$-Zhedanov scheme, which are discussed in the paper.

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arXiv2021

Charting the $q$-Askey scheme

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.

math.CA↗

arXiv2021

Orthogonal polynomials, a short introduction

This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.

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arXiv2021

A nonsymmetric version of Okounkov's BC-type interpolation Macdonald polynomials

Nonsymmetric interpolation Laurent polynomials in $n$ variables are introduced, with the interpolation points depending on $q$ and on a $n$-tuple of parameters $τ=(τ_1,\ldots,τ_n)$. When $τ_i=st^{n-i}$ Okounkov's $3$-parameter $BC_n$-type interpolation Macdonald polynomials are recovered from the nonsymmetric interpolation Laurent polynomials through Hecke algebra symmetrisation with respect to a type $C_n$ Hecke algebra action. In the appendix we give some conjectures about extra vanishing, based on Mathematica computations in rank two.

math.QA↗

arXiv2021

A short survey of duality in special functions

This is a tutorial on duality properties of special functions, mainly of orthogonal polynomials in the ($q$-)Askey scheme. It is based on the first part of the 2017 R.P. Agarwal Memorial Lecture delivered by the author.

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arXiv2021

Dual addition formulas: the case of continuous $q$-ultraspherical and $q$-Hermite polynomials

We settle the dual addition formula for continuous $q$-ultraspherical polynomials as an expansion in terms of special $q$-Racah polynomials for which the constant term is given by the linearization formula for the continuous $q$-ultraspherical polynomials. In a second proof we derive the dual addition formula from the Rahman--Verma addition formula for these polynomials by using the self-duality of the polynomials. We also consider the limit case of continuous $q$-Hermite polynomials.

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arXiv2021

Differentiation by integration using orthogonal polynomials, a survey

This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we unify the continuous and discrete case. We make many side remarks, for instance on wavelets, Mantica's Fourier-Bessel functions and Greville's minimum R_alpha formulas in connection with discrete smoothing.

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arXiv2020

Dual addition formulas associated with dual product formulas

We observe that the linearization coefficients for ultraspherical polynomials are the orthogonality weights for Racah polynomials with special parameters. Then it turns out that the linearization sum with such a Racah polynomial as extra factor inserted, can also be evaluated. The corresponding Fourier--Racah expansion is an addition type formula which is dual to the well-known addition formula for ultraspherical polynomials. The limit to the case of Hermite polynomials of this dual addition formula is also considered. Similar results as for ultraspherical polynomials, although only formal, are given by taking the Ruijsenaars--Hallnäs dual product formula for Gegenbauer functions as a starting point and by working with Wilson polynomials.

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arXiv2020

Integral representations for Horn's $H_2$ function and Olsson's $F_P$ function

We derive some Euler type double integral representations for hypergeometric functions in two variables. In the first part of this paper we deal with Horn's $H_2$ function, in the second part with Olsson's $F_P$ function. Our double integral representing the $F_P$ function is compared with the formula for the same integral representing an $H_2$ function by M. Yoshida (Hiroshima Math. J. 10 (1980), 329-335 and M. Kita (Japan. J. Math. 18 (1992), 25-74). As specified by Kita, their integral is defined by a homological approach. We present a classical double integral version of Kita's integral, with outer integral over a Pochhammer double loop, which we can evaluate as $H_2$ just as Kita did for his integral. Then we show that shrinking of the double loop yields a sum of two double integrals for $F_P$.

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arXiv2018

Dualities in the $q$-Askey scheme and degenerate DAHA

The Askey-Wilson polynomials are a four-parameter family of orthogonal symmetric Laurent polynomials $R_n[z]$ which are eigenfunctions of a second-order $q$-difference operator $L$, and of a second-order difference operator in the variable $n$ with eigenvalue $z +z^{-1}=2x$. Then $L$ and multiplication by $z+z^{-1}$ generate the Askey-Wilson (Zhedanov) algebra. A nice property of the Askey-Wilson polynomials is that the variables $z$ and $n$ occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the non-symmetric case and in the underlying algebraic structures: the Askey-Wilson algebra and the double affine Hecke algebra (DAHA). In this paper we follow the degeneration of the Askey-Wilson polynomials until two arrows down and in four different situations: for the orthogonal polynomials themselves, for the degenerate Askey-Wilson algebras, for the non-symmetric polynomials and for the (degenerate) DAHA and its representations.

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arXiv2018

Quadratic transformations for orthogonal polynomials in one and two variables

We discuss quadratic transformations for orthogonal polynomials in one and two variables. In the one-variable case we list many (or all) quadratic transformations between families in the Askey scheme or $q$-Askey scheme. In the two-variable case we focus, after some generalities, on the polynomials associated with root system $BC_2$, i.e., $BC_2$-type Jacobi polynomials if $q=1$ and Koornwinder polynomials in two variables in the $q$-case.

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