arXiv subjects

Erik Koelink

Publications and source records attributed to Erik Koelink.

At least 19 recordsLinked to original sources

arXiv2025

An evolution of matrix-valued orthogonal polynomials

We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating functions, distribution of zeros for individual entries of the matrices and new type of differential-difference structure. We further speculate about other potentials of the connection formulas found. Part of our proofs makes use of creative telescoping in a matrix setting$-$the strategy which is not yet developed algorithmically.

math.CA↗

arXiv2025

A partial-sum deformation for a family of orthogonal polynomials

There are several questions one may ask about polynomials $q_m(x)=q_m(x;t)=\sum_{n=0}^mt^mp_n(x)$ attached to a family of orthogonal polynomials $\{p_n(x)\}_{n\ge0}$. In this note we draw attention to the naturalness of this partial-sum deformation and related beautiful structures. In particular, we investigate the location and distribution of zeros of $q_m(x;t)$ in the case of varying real parameter $t$.

math.CA↗

arXiv2023

Matrix exceptional Laguerre polynomials

We give an analog of exceptional polynomials in the matrix valued setting by considering suitable factorizations of a given second order differential operator and performing Darboux transformations. Orthogonality and density of the exceptional sequence is discussed in detail. We give an example of matrix valued exceptional Laguerre polynomials of arbitrary size.

math.CA↗

arXiv2022

$BC_2$ type multivariable matrix functions and matrix spherical functions

Matrix spherical functions associated to the compact symmetric pair $(\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(m))$, having reduced root system of type $\mathrm{BC}_2$, are studied. We consider an irreducible $K$-representation $(π,V)$ arising from the $\mathrm{U}(2)$-part of $K$, and the induced representation $\mathrm{Ind}_K^G π$ splits multiplicity free. The corresponding spherical functions, i.e. $Φ\colon G \to \mathrm{End}(V)$ satisfying $Φ(k_1gk_2)=π(k_1)Φ(g)π(k_2)$ for all $g\in G$, $k_1,k_2\in K$, are studied by studying certain leading coefficients which involve hypergeometric functions. This is done explicitly using the action of the radial part of the Casimir operator on these functions and their leading coefficients. To suitably grouped matrix spherical functions we associate two-variable matrix orthogonal polynomials giving a matrix analogue of Koornwinder's 1970s two-variable orthogonal polynomials, which are Heckman-Opdam polynomials for $\mathrm{BC}_2$. In particular, we find explicit orthogonality relations and the matrix polynomials being eigenfunctions to an explicit second order matrix partial differential operator. The scalar part of the matrix weight is less general than Koornwinder's weight.

math.CA↗

arXiv2021

Orthogonal functions related to Lax pairs in Lie algebras

We study a Lax pair in a $2$-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of $L$ and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Moreover, eigenfunctions for the operator $L$ for a Lax pair for $\mathfrak{sl}(d+1,\mathbb C)$ are studied in certain representations.

math.CA↗

arXiv2021

Matrix-valued orthogonal polynomials related to the quantum analogue of $(SU(2) \times SU(2), \text{diag})$

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of $(SU(2) \times SU(2), \text{diag})$ are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey-Wilson polynomials. For these matrix-valued orthogonal polynomials a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, the matrix-valued Askey-Wilson type $q$-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous $q$-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous $q$-ultraspherical polynomials and $q$-Racah polynomials.

math.CA↗

arXiv2019

Non-extremal weight modules for quantized universal enveloping algebras

For quantized universal enveloping algebras we construct weight modules by inducing representations of the centralizer of the Cartan subalgebra in the quantized universal enveloping algebra. The induced modules arising from finite-dimensional weight modules the centralizer algebra are studied. In particular, we study the induction of one-dimensional modules, and this is related to the study of commutative subalgebras of the centralizer algebra. For the special case of $U_q(\mathfrak{sl}(2,\mathbb{C}))$ we show that we get the admissible unitary representations corresponding to the non-compact real form $U_q(\mathfrak{su}(1,1))$.

math.QA↗

arXiv2019

Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra

New convolution identities for orthogonal polynomials belonging to the $q=-1$ analog of the Askey-scheme are obtained. A specialization of the Chihara polynomials will play a central role as the eigenfunctions of a special element of the Lie superalgebra $\mathfrak{osp}(1|2)$ in the positive discrete series representation. Using the Clebsch-Gordan coefficients, a convolution identity for the Specialized Chihara, the dual -1 Hahn and the Big -1 Jacobi polynomials is found. Using the Racah coefficients, a convolution identity for the Big -1 Jacobi and the Bannai-Ito polynomials is found. Finally, these results are applied to construct a bilinear generating function for the Big -1 Jacobi polynomials.

math.CA↗

arXiv2018

Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice

A general family of matrix valued Hermite type orthogonal polynomials is introduced and studied in detail by deriving Pearson equations for the weight and matrix valued differential equations for these matrix polynomials. This is used to derive Rodrigues formulas, explicit formulas for the squared norm and to give an explicit expression of the matrix entries as well to derive a connection formula for the matrix polynomials of Hermite type. We derive matrix valued analogues of Burchnall formulas in operational form as well explicit expansions for the matrix valued Hermite type orthogonal polynomials as well as for previously introduced matrix valued Gegenbauer type orthogonal polynomials. The Burchnall approach gives two descriptions of the matrix valued orthogonal polynomials for the Toda modification of the matrix weight for the Hermite setting. In particular, we obtain a non-trivial solution to the non-abelian Toda lattice equations.

math.CA↗

arXiv2018

Matrix valued Laguerre polynomials

Matrix valued Laguerre polynomials are introduced via a matrix weight function involving several degrees of freedom using the matrix nature. Under suitable conditions on the parameters the matrix weight function satisfies matrix Pearson equations, which allow to introduce shift operators for these polynomials. The shift operators lead to explicit expressions for the structures of these matrix valued Laguerre polynomials, such as a Rodrigues formula, the coefficients in the three-term recurrence, differential operators, and expansion formulas.

math.CA↗

arXiv2018

$SU_q(3)$ corepresentations and bivariate $q$-Krawtchouk polynomials

The matrix elements of unitary $SU_q(3)$ corepresentations, which are analogues of the symmetric powers of the natural repesentation, are shown to be the bivariate $q$-Krawtchouk orthogonal polynomials, thus providing an algebraic interpretation of these polynomials in terms of quantum groups.

math-ph↗

arXiv2018

q-special functions, basic hypergeometric series and operators

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 Hilbert space --a weighted sequence space-- on which this operator is symmetric. Then the corresponding eigenfunctions are polynomials, which are the little $q$-Jacobi polynomials. These polynomials form a family in the $q$-Askey scheme, and so many important properties are well known. In particular, we show how the orthogonality relations and the three-term recurrence for the little $q$-Jacobi polynomials can be obtained using only the factorisation of the corresponding operator. As a next step we consider the $q$-hypergeometric operator in general, which leads to the little $q$-Jacobi functions. We sketch the derivation of the corresponding orthogonality using the connection between various eigenfunctions. The link between the $q$-hypergeometric operators with different parameters is studied in general using $q$-analogues of fractional derivatives, and this gives transmutation properties for this operator. In the final parts of these notes we consider partial extensions of this approach to orthogonal polynomials and special functions. The first extension is a brief introduction to the Askey-Wilson functions and the corresponding integral transform. The second extension is concerned with a matrix-valued extension of the $q$-hypergeometric difference equation and its solutions.

math.CA↗

arXiv2018

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big $q$-Jacobi polynomials and big $q$-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials.

math.CA↗

arXiv2017

Matrix elements of irreducible representations of $\mathrm{SU}(n+1)\times\mathrm{SU}(n+1)$ and multivariable matrix-valued orthogonal polynomials

In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators. In Part 2 we verify that the group case $\mathrm{SU}(n+1)$ meets all the conditions that we impose in Part 1. For any $k\in\mathbb{N}_{0}$ we obtain families of orthogonal polynomials in $n$ variables with values in the $N\times N$-matrices, where $N=\binom{n+k}{k}$. The case $k=0$ leads to the classical Heckman-Opdam polynomials of type $A_{n}$ with geometric parameter. For $k=1$ we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever $n\ge2$. We also give explicit expressions of the spherical functions that determine the matrix weight for $k=1$. These expressions are used to calculate the spherical functions that determine the matrix weight for general $k$ up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case $n=1$. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for $(n,k)$ equal to $(2,1)$ and $(3,1)$.

math.RT↗

arXiv2016

Applications of spectral theory to special functions

Lecture notes for one of the courses at the OPSFA Summerschool 6, July 11-15, 2016. All the results in these notes have appeared in the literature. Many special functions are eigenfunctions to explicit operators, such as difference and differential operators, which is in particular true for the special functions occurring in the Askey-scheme, its $q$-analogue and extensions. The study of the spectral properties of such operators leads to explicit information for the corresponding special functions. We discuss several instances of this application, involving orthogonal polynomials and their matrix-valued analogues.

math.CA↗

arXiv2016

Matrix-valued Gegenbauer polynomials

We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter $ν>0$. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameter $ν$ and $ν+1$. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials are the matrix-valued Gegenbauer polynomials which are eigenfunctions for the symmetric matrix-valued differential operators. Using the shift operators we find the squared norm and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit non-trivial expression for the matrix entries of the matrix-valued Gegenbauer polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case $ν=1$ reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.

math.CA↗

arXiv2016

Branching rules for finite-dimensional $\mathcal{U}_q(\mathfrak{su}(3))$-representations with respect to a right coideal subalgebra

We consider the quantum symmetric pair $(\mathcal{U}_q(\mathfrak{su}(3)), \mathcal{B})$ where $\mathcal{B}$ is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of $\mathcal{B}$ are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of $\mathcal{U}_q(\mathfrak{su}(3))$ to $\mathcal{B}$ decomposes multiplicity free into irreducible representations of $\mathcal{B}$. Furthermore we give explicit expressions for the highest weight vectors in this decomposition in terms of dual $q$-Krawtchouk polynomials.

math.RT↗

arXiv2016

Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures

A matrix-valued measure $Θ$ reduces to measures of smaller size if there exists a constant invertible matrix $M$ such that $MΘM^*$ is block diagonal. Equivalently, the real vector space ${\mathscr A}$ of all matrices $T$ such that $TΘ(X)=Θ(X) T^*$ for any Borel set $X$ is non-trivial. If the subspace $A_h$ of self-adjoints elements in the commutant algebra $A$ of $Θ$ is non-trivial, then $Θ$ is reducible via a unitary matrix. In this paper we prove that ${\mathscr A}$ is $*$-invariant if and only if $A_h={\mathscr A}$, i.e., every reduction of $Θ$ can be performed via a unitary matrix. The motivation for this paper comes from families of matrix-valued polynomials related to the group ${\rm SU}(2)\times {\rm SU}(2)$ and its quantum analogue. In both cases the commutant algebra $A=A_h\oplus iA_h$ is of dimension two and the matrix-valued measures reduce unitarily into a $2\times 2$ block diagonal matrix. Here we show that there is no further non-unitary reduction.

math.CA↗