Page 201 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 201
ORTHOGONAL POLYNOMIALS AND SPECIAL FUNCTIONS (MS-10)
local asymptotics was introduced for Legendre orthogonal polynomials (OP) by the German
mathematicians H. E. Heine and G. F. Mehler in the 19th century. Later, it was extended to the
families of classical OP (Jacobi, Laguerre, Hermite), and more recently, these formulae were
obtained for other families as discrete OP, generalized Freud OP, multiple OP or Sobolev OP,
among others.
These formulae have a nice consequence about the scaled zeros of the polynomials, i.e.
using the well–known Hurwitz’s theorem we can establish a limit relation between these scaled
zeros and the ones of a Bessel function of the first kind. In this way, we are looking for a similar
result in the context of the q-analysis and we will illustrate the results with numerical examples.
A proof of a conjecture of Elbert and Laforgia on the zeros of cylinder
functions
Gergo˝ Nemes, nemes.gergo@renyi.hu
Alfréd Rényi Institute of Mathematics, Hungary
We prove the enveloping property of the known divergent asymptotic expansion of the large real
zeros of the cylinder functions, and thereby answering in the affirmative a conjecture posed by
Elbert and Laforgia in 2001 (J. Comput. Appl. Math. 133 (2001), no. 1–2, p. 683). The essence
of the proof is the construction of an analytic function that returns the zeros when evaluated
along certain discrete sets of real numbers. By manipulating contour integrals of this function,
we derive the asymptotic expansion of the large zeros truncated after a finite number of terms
plus a remainder that can be estimated efficiently. The conjecture is then deduced as a corollary
of this estimate.
Multivariate hybrid orthogonal functions
Teresa E. Perez, tperez@ugr.es
Dep. de Matemática Aplicada & Math Institute IEMath-GR,
Universidad de Granada, Spain
Coauthor: Cleonice F. Bracciali
We consider multivariate orthogonal functions satisfying hybrid orthogonality conditions with
respect to a moment functional. This kind of orthogonality means that the product of func-
tions of different parity order is computed by means of the moment functional, and the product
of elements of the same parity order is computed by a modification of the original moment
functional. Results about existence conditions, three term relations with matrix coefficients,
a Favard type theorem for this kind of hybrid orthogonal functions are proved. In addition, a
method to construct bivariate hybrid orthogonal functions from univariate orthogonal polyno-
mials and univariate orthogonal functions is presented. Finally, we give a complete description
of a sequence of hybrid orthogonal functions on the unit disk on R2, that includes, as particular
case, the classical orthogonal polynomials on the disk.
199
local asymptotics was introduced for Legendre orthogonal polynomials (OP) by the German
mathematicians H. E. Heine and G. F. Mehler in the 19th century. Later, it was extended to the
families of classical OP (Jacobi, Laguerre, Hermite), and more recently, these formulae were
obtained for other families as discrete OP, generalized Freud OP, multiple OP or Sobolev OP,
among others.
These formulae have a nice consequence about the scaled zeros of the polynomials, i.e.
using the well–known Hurwitz’s theorem we can establish a limit relation between these scaled
zeros and the ones of a Bessel function of the first kind. In this way, we are looking for a similar
result in the context of the q-analysis and we will illustrate the results with numerical examples.
A proof of a conjecture of Elbert and Laforgia on the zeros of cylinder
functions
Gergo˝ Nemes, nemes.gergo@renyi.hu
Alfréd Rényi Institute of Mathematics, Hungary
We prove the enveloping property of the known divergent asymptotic expansion of the large real
zeros of the cylinder functions, and thereby answering in the affirmative a conjecture posed by
Elbert and Laforgia in 2001 (J. Comput. Appl. Math. 133 (2001), no. 1–2, p. 683). The essence
of the proof is the construction of an analytic function that returns the zeros when evaluated
along certain discrete sets of real numbers. By manipulating contour integrals of this function,
we derive the asymptotic expansion of the large zeros truncated after a finite number of terms
plus a remainder that can be estimated efficiently. The conjecture is then deduced as a corollary
of this estimate.
Multivariate hybrid orthogonal functions
Teresa E. Perez, tperez@ugr.es
Dep. de Matemática Aplicada & Math Institute IEMath-GR,
Universidad de Granada, Spain
Coauthor: Cleonice F. Bracciali
We consider multivariate orthogonal functions satisfying hybrid orthogonality conditions with
respect to a moment functional. This kind of orthogonality means that the product of func-
tions of different parity order is computed by means of the moment functional, and the product
of elements of the same parity order is computed by a modification of the original moment
functional. Results about existence conditions, three term relations with matrix coefficients,
a Favard type theorem for this kind of hybrid orthogonal functions are proved. In addition, a
method to construct bivariate hybrid orthogonal functions from univariate orthogonal polyno-
mials and univariate orthogonal functions is presented. Finally, we give a complete description
of a sequence of hybrid orthogonal functions on the unit disk on R2, that includes, as particular
case, the classical orthogonal polynomials on the disk.
199
