Page 202 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 202
ORTHOGONAL POLYNOMIALS AND SPECIAL FUNCTIONS (MS-10)
Comparative asymptotics of rational modified orthogonal polynomials
Hector Pijeira Cabrera, hpijeira@math.uc3m.es
Universidad Carlos III de Madrid, Spain
Coauthor: Javier Quintero-Roba
The talk deals with the relative asymptotic behavior between a sequence of orthogonal polyno-
mials with respect to a measure with unbounded support and the orthogonal polynomials with
respect to a rational modification of this measure. These results are applied to the study of the
asymptotic behavior of Sobolev type orthogonal polynomials.
Converting divergent asymptotic series into convergent series: factorial
series for Laplace-type integrals
Ester Pérez Sinusía, ester.perez@unizar.es
University of Zaragoza and IUMA, Spain
Coauthors: Chelo Ferreira, José Luis López
Asymptotic techniques for Laplace-type integrals are a useful tool to derive asymptotic approx-
imations of special functions. But in most of the important examples of special functions, the
asymptotic expansion is not convergent. In this paper we investigate a modification of those
asymptotic techniques that transforms the unbounded integration region of the Laplace-type in-
tegral into a bounded region. Then, an elementary asymptotic analysis of the new integral shows
that the asymptotically relevant integration point is attained at a point of the boundary of the
integration region and an expansion of the integrand at that point gives an asymptotic expansion
of the integral. But moreover, an analysis of the remainder of this new expansion shows that
it is convergent under a mild condition over the integrand. We illustrate this modification with
several examples of special functions, providing convergent and asymptotic expansions of these
functions.
Khrushchev formulas for orthogonal polynomials
Luis Velázquez, velazque@unizar.es
University of Zaragoza and IUMA, Spain
Coauthors: Christopher Cedzich, F. Alberto Grünbaum, Albert H. Werner, Reinhard F. Werner
Since their origin in the early 20th century, the theory of orthogonal polynomials on the unit cir-
cle (OPUC) has proved to be intimately connected to harmonic analysis via the so called Schur
functions. The beginning of this century has witnessed a renewed interest in this connection
due to a revolutionary approach to OPUC by the hand of Sergei Khrushchev, which emphasizes
the role of continued fractions and Schur functions. The name “Khrushchev theory”, coined by
Barry Simon, refers to a body of methods and results on OPUC originated by this new approach,
whose cornerstone is the so called “Khrushchev formula”. The interest of Schur functions and
Khrushchev formulas has been fueled even more by a recently uncovered link between OPUC
theory and the study of quantum walks, the quantum version of random walks, where Schur
functions are central to develop the quantum version of Pólya’s renewal theory. This has led to
a very general understanding of Khrushchev formula, susceptible to be applied to a wide range
200
Comparative asymptotics of rational modified orthogonal polynomials
Hector Pijeira Cabrera, hpijeira@math.uc3m.es
Universidad Carlos III de Madrid, Spain
Coauthor: Javier Quintero-Roba
The talk deals with the relative asymptotic behavior between a sequence of orthogonal polyno-
mials with respect to a measure with unbounded support and the orthogonal polynomials with
respect to a rational modification of this measure. These results are applied to the study of the
asymptotic behavior of Sobolev type orthogonal polynomials.
Converting divergent asymptotic series into convergent series: factorial
series for Laplace-type integrals
Ester Pérez Sinusía, ester.perez@unizar.es
University of Zaragoza and IUMA, Spain
Coauthors: Chelo Ferreira, José Luis López
Asymptotic techniques for Laplace-type integrals are a useful tool to derive asymptotic approx-
imations of special functions. But in most of the important examples of special functions, the
asymptotic expansion is not convergent. In this paper we investigate a modification of those
asymptotic techniques that transforms the unbounded integration region of the Laplace-type in-
tegral into a bounded region. Then, an elementary asymptotic analysis of the new integral shows
that the asymptotically relevant integration point is attained at a point of the boundary of the
integration region and an expansion of the integrand at that point gives an asymptotic expansion
of the integral. But moreover, an analysis of the remainder of this new expansion shows that
it is convergent under a mild condition over the integrand. We illustrate this modification with
several examples of special functions, providing convergent and asymptotic expansions of these
functions.
Khrushchev formulas for orthogonal polynomials
Luis Velázquez, velazque@unizar.es
University of Zaragoza and IUMA, Spain
Coauthors: Christopher Cedzich, F. Alberto Grünbaum, Albert H. Werner, Reinhard F. Werner
Since their origin in the early 20th century, the theory of orthogonal polynomials on the unit cir-
cle (OPUC) has proved to be intimately connected to harmonic analysis via the so called Schur
functions. The beginning of this century has witnessed a renewed interest in this connection
due to a revolutionary approach to OPUC by the hand of Sergei Khrushchev, which emphasizes
the role of continued fractions and Schur functions. The name “Khrushchev theory”, coined by
Barry Simon, refers to a body of methods and results on OPUC originated by this new approach,
whose cornerstone is the so called “Khrushchev formula”. The interest of Schur functions and
Khrushchev formulas has been fueled even more by a recently uncovered link between OPUC
theory and the study of quantum walks, the quantum version of random walks, where Schur
functions are central to develop the quantum version of Pólya’s renewal theory. This has led to
a very general understanding of Khrushchev formula, susceptible to be applied to a wide range
200
