Page 198 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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ORTHOGONAL POLYNOMIALS AND SPECIAL FUNCTIONS (MS-10)
On some positive quadrature rules on the unit circle
Cleonice F. Bracciali, cleonice.bracciali@unesp.br
UNESP - Universidade Estadual Paulista, Brazil
We consider quadrature rules on the real line associated with a sequence of polynomials gener-
ated by a special RII recurrence relation. With a simple transformation, these quadrature rules
on the real line also lead to certain positive quadrature rules of highest algebraic degree of pre-
cision on the unit circle. We also show new approaches to evaluate the nodes and weights of
these specific quadrature rules on the unit circle.
Joint work with Junior A. Pereira and A. Sri Ranga
Multiple Orthogonal Polynomials and Random Walks
Ana Foulquié, foulquie@ua.pt
University of Aveiro, Portugal
Coauthors: Amílcar Branquinho, Manuel Mañas, Carlos Álvarez-Fernández,
Juan Fernández-Díaz
Given a non-negative Jacobi matrix describing higher order recurrence relations for multiple
orthogonal polynomials of type II and corresponding linear forms of type I, a general strategy
for constructing a pair of stochastic matrices, dual to each other, is provided. The corresponding
Markov chains (or 1D random walks) allow, in one transition, to reach for the N -th previous
states, to remain in the state or reach for the immediately next state. The dual Markov chains
allow, in one transition, to reach for the N -th next states, to remain in the state or reach for im-
mediately previous state. The connection between both dual Markov chains is discussed at the
light of the Poincaré’s theorem on ratio asymptotics for homogeneous linear recurrence relations
and the Christoffel–Darboux formula within the sequence of multiple orthogonal polynomials
and linear forms of type I.
The Karlin–McGregor representation formula is extended to both dual random walks, and
applied to the discussion of the corresponding generating functions and first-passage distribu-
tions. Recurrent or transient character of the Markov chain is discussed. Steady state and some
conjectures on its existence and the relation with mass points are also given.
The Jacobi–Piñeiro multiple orthogonal polynomials are taken as a case study of the de-
scribed results.
d-orthogonal analogs of classical orthogonal polynomials
Emil Horozov, horozov@fmi.uni-sofia.bg
retired, Bulgaria
Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre and Bessel have the
property that the polynomials of each system are eigenfunctions of second order ordinary dif-
ferential operator. According to a classical theorem by Bochner they are the only systems with
this property. The othogonality property is equivalent to the 3-term recurrence relation accord-
ing to the famous Favard-Shohat theorem.
Motivated by Bochner’s theorem we are looking for d-orthogonal polynomials (systems that
196
On some positive quadrature rules on the unit circle
Cleonice F. Bracciali, cleonice.bracciali@unesp.br
UNESP - Universidade Estadual Paulista, Brazil
We consider quadrature rules on the real line associated with a sequence of polynomials gener-
ated by a special RII recurrence relation. With a simple transformation, these quadrature rules
on the real line also lead to certain positive quadrature rules of highest algebraic degree of pre-
cision on the unit circle. We also show new approaches to evaluate the nodes and weights of
these specific quadrature rules on the unit circle.
Joint work with Junior A. Pereira and A. Sri Ranga
Multiple Orthogonal Polynomials and Random Walks
Ana Foulquié, foulquie@ua.pt
University of Aveiro, Portugal
Coauthors: Amílcar Branquinho, Manuel Mañas, Carlos Álvarez-Fernández,
Juan Fernández-Díaz
Given a non-negative Jacobi matrix describing higher order recurrence relations for multiple
orthogonal polynomials of type II and corresponding linear forms of type I, a general strategy
for constructing a pair of stochastic matrices, dual to each other, is provided. The corresponding
Markov chains (or 1D random walks) allow, in one transition, to reach for the N -th previous
states, to remain in the state or reach for the immediately next state. The dual Markov chains
allow, in one transition, to reach for the N -th next states, to remain in the state or reach for im-
mediately previous state. The connection between both dual Markov chains is discussed at the
light of the Poincaré’s theorem on ratio asymptotics for homogeneous linear recurrence relations
and the Christoffel–Darboux formula within the sequence of multiple orthogonal polynomials
and linear forms of type I.
The Karlin–McGregor representation formula is extended to both dual random walks, and
applied to the discussion of the corresponding generating functions and first-passage distribu-
tions. Recurrent or transient character of the Markov chain is discussed. Steady state and some
conjectures on its existence and the relation with mass points are also given.
The Jacobi–Piñeiro multiple orthogonal polynomials are taken as a case study of the de-
scribed results.
d-orthogonal analogs of classical orthogonal polynomials
Emil Horozov, horozov@fmi.uni-sofia.bg
retired, Bulgaria
Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre and Bessel have the
property that the polynomials of each system are eigenfunctions of second order ordinary dif-
ferential operator. According to a classical theorem by Bochner they are the only systems with
this property. The othogonality property is equivalent to the 3-term recurrence relation accord-
ing to the famous Favard-Shohat theorem.
Motivated by Bochner’s theorem we are looking for d-orthogonal polynomials (systems that
196
