Page 199 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 199
ORTHOGONAL POLYNOMIALS AND SPECIAL FUNCTIONS (MS-10)
satisfy d+2-term recurrence) that are also eigenfunctions of a differential operator. We call these
simultaneous conditions Bochner’s property.
Using purely algebraic methods and ideas from the bispectral problem, based on automor-
phisms of non-commutative algebras we construct polynomial systems with Bochner property.
Many properties of the constructed polynomial systems are obtained quite directly from
their construction. In particular they have a number of similarities with the classical orthogonal
polynomials, which makes them their natural analog - they have hypergeometric representa-
tions, ladder operators, generating functions, they can be presented via Rodrigues formulas,
there are Pearson’s equations for the weights of their measures, they possess the Hahn’s prop-
erty, i.e. the polynomial system of their derivatives are again orthogonal polynomials, etc.
We conjecture that the proposed construction exhausts all systems with Bochner’s property.
Connections to integrable systems like KP and Toda hierarchies and matrix models will be
discussed.
Matrix valued multivariable orthogonal polynomials with BC2-symmetry
Erik Koelink, e.koelink@math.ru.nl
Radboud Universiteit, Netherlands
Coauthor: Jie Liu
The relation between multivariable special functions and orthogonal polynomials has been in-
fluenced greatly by generalising spherical functions on Riemannian symmetric spaces by Heck-
man and Opdam. These functions and polynomials turn out to be important in mathematical
physics. In recent years several approaches to vector and matrix valued analogues in the rank
one case have been studied from the perspective of matrix valued spherical functions, and sev-
eral extensions have been studied. We discuss a rank two case with BC2-symmetry in detail,
and we derive explicit results for the corresponding matrix valued orthogonal polynomials. In
particular, the matrix weight function is given explicitly in terms of Krawtchouk polynomials,
and we prsent the matrix valued linear partial differential operator for which the polynomials
are eigenfunctions.
Periodic random tiling models and non-Hermitian orthogonality
Arno Kuijlaars, arno.kuijlaars@kuleuven.be
Katholieke Universiteit Leuven, Belgium
Certain random tiling models show characteristic features that are similar to the eigenvalues of
large random matrices. They can be recast as determinantal point process from which limiting
laws as the Tracy-Widom distribution at the edge and the Gaussian free field in the bulk can be
deduced.
I will discuss a new technique, developed in collaboration with Maurice Duits, to study
tiling models with periodic weights. The technique relies on a formulation of a correlation
kernel as a double contour integral containing non-Hermitian orthogonal polynomials. In the
case of periodic weightings, the orthogonal polynomials are matrix valued.
We use the Riemann-Hilbert problem for matrix valued orthogonal polynomials to obtain
asymptotics for the two-periodic Aztec diamond. This model is remarkable since it gives rise
to a gaseous phase, in addition to the more familiar solid and liquid phases.
197
satisfy d+2-term recurrence) that are also eigenfunctions of a differential operator. We call these
simultaneous conditions Bochner’s property.
Using purely algebraic methods and ideas from the bispectral problem, based on automor-
phisms of non-commutative algebras we construct polynomial systems with Bochner property.
Many properties of the constructed polynomial systems are obtained quite directly from
their construction. In particular they have a number of similarities with the classical orthogonal
polynomials, which makes them their natural analog - they have hypergeometric representa-
tions, ladder operators, generating functions, they can be presented via Rodrigues formulas,
there are Pearson’s equations for the weights of their measures, they possess the Hahn’s prop-
erty, i.e. the polynomial system of their derivatives are again orthogonal polynomials, etc.
We conjecture that the proposed construction exhausts all systems with Bochner’s property.
Connections to integrable systems like KP and Toda hierarchies and matrix models will be
discussed.
Matrix valued multivariable orthogonal polynomials with BC2-symmetry
Erik Koelink, e.koelink@math.ru.nl
Radboud Universiteit, Netherlands
Coauthor: Jie Liu
The relation between multivariable special functions and orthogonal polynomials has been in-
fluenced greatly by generalising spherical functions on Riemannian symmetric spaces by Heck-
man and Opdam. These functions and polynomials turn out to be important in mathematical
physics. In recent years several approaches to vector and matrix valued analogues in the rank
one case have been studied from the perspective of matrix valued spherical functions, and sev-
eral extensions have been studied. We discuss a rank two case with BC2-symmetry in detail,
and we derive explicit results for the corresponding matrix valued orthogonal polynomials. In
particular, the matrix weight function is given explicitly in terms of Krawtchouk polynomials,
and we prsent the matrix valued linear partial differential operator for which the polynomials
are eigenfunctions.
Periodic random tiling models and non-Hermitian orthogonality
Arno Kuijlaars, arno.kuijlaars@kuleuven.be
Katholieke Universiteit Leuven, Belgium
Certain random tiling models show characteristic features that are similar to the eigenvalues of
large random matrices. They can be recast as determinantal point process from which limiting
laws as the Tracy-Widom distribution at the edge and the Gaussian free field in the bulk can be
deduced.
I will discuss a new technique, developed in collaboration with Maurice Duits, to study
tiling models with periodic weights. The technique relies on a formulation of a correlation
kernel as a double contour integral containing non-Hermitian orthogonal polynomials. In the
case of periodic weightings, the orthogonal polynomials are matrix valued.
We use the Riemann-Hilbert problem for matrix valued orthogonal polynomials to obtain
asymptotics for the two-periodic Aztec diamond. This model is remarkable since it gives rise
to a gaseous phase, in addition to the more familiar solid and liquid phases.
197
