arXiv subjects

Carel Wagenaar

Publications and source records attributed to Carel Wagenaar.

6 recordsLinked to original sources

arXiv2026

The two-weight fractional Poincaré-Sobolev sandwich

We establish a two-weight fractional Poincaré-Sobolev sandwich, consisting of a two-weight fractional Poincaré-Sobolev inequality and a two-weight embedding from the first-order Sobolev space to a Triebel-Lizorkin space defined via a difference norm. Our constants are asymptotically sharp as the fractional parameter approaches $1$. Our results are new even in the one-weight case. For each inequality we give explicit quantitative dependence on Muckenhoupt weight characteristics and treat both subcritical and critical regimes, the former via elementary methods and the latter via sparse domination. As one of our main tools, we establish a new sparse domination result for Triebel-Lizorkin difference norms. Our methods unify, simplify and significantly extend various earlier approaches.

math.CA↗

arXiv2026

Bispectral rational functions and Leonard trios

It is well-known that Leonard pairs have a close connection with bispectral orthogonal polynomials of the Askey scheme. In this paper, we introduce the notion of a Leonard trio $(V,\oV,Z)$, an algebraic structure extending Leonard pairs, for which the overlap coefficients of eigenfunctions of $V$ and $\oV$ are biorthogonal rational functions satisfying generalized eigenvalue problems. We introduce and start the classification of irreducible Leonard trios by using its connection with Leonard pairs and Heun operators. In particular, we show that Wilson's rational functions appear as overlap coefficients, prove its difference, recurrence and biorthogonality relations, and obtain a summation formula expressing them as a finite sum of products of two $q$-Racah polynomials. We also begin to investigate reduced Leonard trios, for which the general eigenvalue problem simplifies to a $R_I$-type recurrence relation. As an illustration, we present an example of this in which the rational functions appearing as overlap coefficients can be expressed as a ${}_{4}ϕ_3$ and are associated with a Leonard pair of dual $q$-Hahn type.

math.RA↗

arXiv2025

Quantum algebra approach to univariate and multivariate rational functions of $q$-Racah type

In this paper, we study rational functions of $q$-Racah type and a multivariate extension, using representation theory of $\mathcal U_q(\mathfrak{sl}_2)$. Eigenfunctions of twisted primitive elements in $\mathcal U_q(\mathfrak{su}_2)$ can be expressed in terms of $q^{-1}$-Krawtchouk polynomials. Using this, we show that overlap coefficients of solutions of a generalized eigenvalue problem (GEVP) and an eigenvalue problem (EVP) can be expressed in terms of a rational function of $_4φ_3$-type. With help of the quantum algebra, we derive (bi)orthogonality relations as well as a GEVP for these functions. Furthermore, using this new algebraic interpretation, we can exploit the co-algebra structure of $\mathcal U_q(\mathfrak{sl}_2)$ to find a multivariate extension of these rational functions and derive biorthogonality relations and GEVPs for the multivariate functions. Then we repeat this procedure for the non-compact quantum algebra $\mathcal U_q(\mathfrak{su}_{1,1})$, where the $q^{-1}$-Al-Salam--Chihara polynomials play the role of the $q^{-1}$-Krawtchouk polynomials. As an application of the multivariate rational functions, we show that they appear as duality functions for certain interacting particle systems.

math.QA↗

arXiv2025

Dynamic generalizations of the Asymmetric Inclusion Process, Asymmetric Brownian Energy Process and their Dualities

Two new interacting particle systems are introduced in this paper: dynamic versions of the asymmetric inclusion process (ASIP) and the asymmetric Brownian energy process (ABEP). Dualities and reversibility of these processes are proven, where the quantum algebra $\mathcal{U}_q(\mathfrak{su}(1,1))$ and the Al-Salam--Chihara polynomials play a crucial role. Two hierarchies of duality functions are found, where the Askey-Wilson polynomials and Jacobi polynomials sit on top.

math.PR↗

arXiv2024

A Generalized Dynamic Asymmetric Exclusion Process: Orthogonal Dualities and Degenerations

In this paper, a generalized version of dynamic ASEP is introduced, and it is shown that the process has a Markov duality property with the same process on the reversed lattice. The duality functions are multivariate $q$-Racah polynomials, and the corresponding orthogonality measure is the reversible measure of the process. By taking limits in the generator of dynamic ASEP, its reversible measure, and the duality functions, we obtain orthogonal and triangular dualities for several other interacting particle systems. In this sense, the duality of dynamic ASEP sits on top of a hierarchy of many dualities. For the construction of the process, we rely on representation theory of the quantum algebra $\mathcal{U}_q(\mathfrak{sl}_2)$. In the standard representation, the generator of generalized ASEP can be constructed from the coproduct of the Casimir. After a suitable change of representation, we obtain the generator of dynamic ASEP. The corresponding intertwiner is constructed from $q$-Krawtchouk polynomials, which arise as eigenfunctions of twisted primitive elements. This gives a duality between dynamic ASEP and generalized ASEP with $q$-Krawtchouk polynomials as duality functions. Using this duality, we show the (almost) self-duality of dynamic ASEP.

math.PR↗

arXiv2023

An Askey-Wilson Algebra of Rank 2

An algebra is introduced which can be considered as a rank 2 extension of the Askey-Wilson algebra. Relations in this algebra are motivated by relations between coproducts of twisted primitive elements in the two-fold tensor product of the quantum algebra $\mathcal{U}_{q}(\mathfrak{sl}(2,\mathbb C))$. It is shown that bivariate $q$-Racah polynomials appear as overlap coefficients of eigenvectors of generators of the algebra. Furthermore, the corresponding $q$-difference operators are calculated using the defining relations of the algebra, showing that it encodes the bispectral properties of the bivariate $q$-Racah polynomials.

math.QA↗