Page 200 - 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)
Bivariate Koornwinder-Sobolev orthogonal polynomials
Misael Enrique Marriaga Castillo, misael.marriaga@urjc.es
Universidad Rey Juan Carlos, Spain
Coauthors: Miguel Piñar, Teresa E. Pérez
The purpose of this talk is to introduce the so-called Koornwinder bivariate orthogonal poly-
nomials. These polynomials are generated by means of a non-trivial procedure involving two
families of univariate orthogonal polynomials and a function ρ(t) such that ρ(t)2 is a polynomial
of degree less than or equal to 2. We also discuss how to extend the Koornwinder method to the
case when one of the univariate families is orthogonal with respect to a Sobolev inner product.
Therefore, we study the new Sobolev bivariate families obtaining relations between the clas-
sical original Koornwinder polynomials and the Sobolev one, deducing recursive methods in
order to compute the coefficients. The case when one of the univariate families is associated to
a classical inner product is analysed. Finally, some useful examples are given.
Poncelet’s Theorem and Orthogonal Polynomials
Andrei Martinez Finkelshtein, a_martinez-finkelshtein@baylor.edu
Baylor University and Universidad de Almeria, Spain
Poncelet’s Theorem is one of the most beautiful and well known results from projective geom-
etry. In the last few decades, the relationship between Poncelet’s Theorem and other mathe-
matical object, such as Blaschke products or numerical range of completely non-unitary con-
tractions, has been the focus of extensive research. Recently, another connection, now with the
theory of orthogonal polynomials on the unit circle has been revealed. These interconnections
allow us to prove several new results, to interpret the existing theory in a new context, and also
to understand further connections with other areas of geometry and analysis.
This is a joint work with M. Hunziker, T. Poe, and B. Simanek.
Local asymptotics for some q-hypergeometric polynomials
Juan F. Mañas Mañas, jmm939@ual.es
Universidad de Almería, Spain
Coauthor: Juan J. Moreno-Balcázar
The basic q-hypergeometric function rφs is defined by the series
r φs a1, . . . , ar ; q, z = ∞ (a1; q)k · · · (ar; q)k (−1)kq(k2) 1+s−r zk (1)
b1, . . . , bs k=0 (b1; q)k · · · (bs; q)k ,
(q; q)k
where 0 < q < 1 and (aj; q)k and (bj; q)k denote the q-analogues of the Pochhammer symbol.
When one of the parameters aj in (1) is equal to q−n the basic q-hypergeo-metric function
is a polynomial of degree at most n in the variable z. Our objective is to obtain a type of local
asymptotics, known as Mehler–Heine asymptotics, for q-hypergeometric polynomials when
r = s.
Concretely, by scaling adequately these polynomials we intend to get a limit relation be-
tween them and a q-analogue of the Bessel function of the first kind. Originally, this type of
198
Bivariate Koornwinder-Sobolev orthogonal polynomials
Misael Enrique Marriaga Castillo, misael.marriaga@urjc.es
Universidad Rey Juan Carlos, Spain
Coauthors: Miguel Piñar, Teresa E. Pérez
The purpose of this talk is to introduce the so-called Koornwinder bivariate orthogonal poly-
nomials. These polynomials are generated by means of a non-trivial procedure involving two
families of univariate orthogonal polynomials and a function ρ(t) such that ρ(t)2 is a polynomial
of degree less than or equal to 2. We also discuss how to extend the Koornwinder method to the
case when one of the univariate families is orthogonal with respect to a Sobolev inner product.
Therefore, we study the new Sobolev bivariate families obtaining relations between the clas-
sical original Koornwinder polynomials and the Sobolev one, deducing recursive methods in
order to compute the coefficients. The case when one of the univariate families is associated to
a classical inner product is analysed. Finally, some useful examples are given.
Poncelet’s Theorem and Orthogonal Polynomials
Andrei Martinez Finkelshtein, a_martinez-finkelshtein@baylor.edu
Baylor University and Universidad de Almeria, Spain
Poncelet’s Theorem is one of the most beautiful and well known results from projective geom-
etry. In the last few decades, the relationship between Poncelet’s Theorem and other mathe-
matical object, such as Blaschke products or numerical range of completely non-unitary con-
tractions, has been the focus of extensive research. Recently, another connection, now with the
theory of orthogonal polynomials on the unit circle has been revealed. These interconnections
allow us to prove several new results, to interpret the existing theory in a new context, and also
to understand further connections with other areas of geometry and analysis.
This is a joint work with M. Hunziker, T. Poe, and B. Simanek.
Local asymptotics for some q-hypergeometric polynomials
Juan F. Mañas Mañas, jmm939@ual.es
Universidad de Almería, Spain
Coauthor: Juan J. Moreno-Balcázar
The basic q-hypergeometric function rφs is defined by the series
r φs a1, . . . , ar ; q, z = ∞ (a1; q)k · · · (ar; q)k (−1)kq(k2) 1+s−r zk (1)
b1, . . . , bs k=0 (b1; q)k · · · (bs; q)k ,
(q; q)k
where 0 < q < 1 and (aj; q)k and (bj; q)k denote the q-analogues of the Pochhammer symbol.
When one of the parameters aj in (1) is equal to q−n the basic q-hypergeo-metric function
is a polynomial of degree at most n in the variable z. Our objective is to obtain a type of local
asymptotics, known as Mehler–Heine asymptotics, for q-hypergeometric polynomials when
r = s.
Concretely, by scaling adequately these polynomials we intend to get a limit relation be-
tween them and a q-analogue of the Bessel function of the first kind. Originally, this type of
198
