Dissertation
On Connection, Linearization and Duplication Coefficients of Classical Orthogonal Polynomials
Abstract
In this work, we have mainly achieved the following:
1. we provide a review of the main methods used for the computation of the connection and linearization coefficients between orthogonal polynomials of a continuous variable, moreover using a new approach, the duplication problem of these polynomial families is solved;
2. we review the main methods used for the computation of the connection and linearization coefficients of orthogonal polynomials of a discrete variable, we solve the duplication and linearization problem of all orthogonal polynomials of a discrete variable;
3. we propose a method to generate the connection, linearization and duplication coefficients for q-orthogonal polynomials;
4. we propose a unified method to obtain these coefficients in a generic way for orthogonal polynomials on quadratic and q-quadratic lattices.
Our algorithmic approach to compute linearization, connection and duplication coefficients is based on the one used by Koepf and Schmersau and on the NaViMa algorithm. Our main technique is to use explicit formulas for structural identities of classical orthogonal polynomial systems. We find our results by an application of computer algebra. The major algorithmic tools for our development are Zeilberger’s algorithm, q-Zeilberger’s algorithm, the Petkovšek-van-Hoeij algorithm, the q-Petkovšek-van-Hoeij algorithm, and Algorithm 2.2, p. 20 of Koepf's book "Hypergeometric Summation" and it q-analogue.
1. we provide a review of the main methods used for the computation of the connection and linearization coefficients between orthogonal polynomials of a continuous variable, moreover using a new approach, the duplication problem of these polynomial families is solved;
2. we review the main methods used for the computation of the connection and linearization coefficients of orthogonal polynomials of a discrete variable, we solve the duplication and linearization problem of all orthogonal polynomials of a discrete variable;
3. we propose a method to generate the connection, linearization and duplication coefficients for q-orthogonal polynomials;
4. we propose a unified method to obtain these coefficients in a generic way for orthogonal polynomials on quadratic and q-quadratic lattices.
Our algorithmic approach to compute linearization, connection and duplication coefficients is based on the one used by Koepf and Schmersau and on the NaViMa algorithm. Our main technique is to use explicit formulas for structural identities of classical orthogonal polynomial systems. We find our results by an application of computer algebra. The major algorithmic tools for our development are Zeilberger’s algorithm, q-Zeilberger’s algorithm, the Petkovšek-van-Hoeij algorithm, the q-Petkovšek-van-Hoeij algorithm, and Algorithm 2.2, p. 20 of Koepf's book "Hypergeometric Summation" and it q-analogue.
Citation
@phdthesis{urn:nbn:de:hebis:34-2014071645714,
author={Tcheutia, Daniel Duviol},
title={On Connection, Linearization and Duplication Coefficients of Classical Orthogonal Polynomials},
school={Universität, Kassel, Fachbereich Mathematik und Naturwissenschaften},
month={07},
year={2014}
}
0500 Oax 0501 Text $btxt$2rdacontent 0502 Computermedien $bc$2rdacarrier 1100 2014$n2014 1500 1/eng 2050 ##0##urn:nbn:de:hebis:34-2014071645714 3000 Tcheutia, Daniel Duviol 4000 On Connection, Linearization and Duplication Coefficients of Classical Orthogonal Polynomials / Tcheutia, Daniel Duviol 4030 4060 Online-Ressource 4085 ##0##=u http://nbn-resolving.de/urn:nbn:de:hebis:34-2014071645714=x R 4204 \$dDissertation 4170 5550 {{Orthogonale Polynome}} 7136 ##0##urn:nbn:de:hebis:34-2014071645714
2014-07-16T10:47:08Z 2014-07-16T10:47:08Z 2014-07-16 urn:nbn:de:hebis:34-2014071645714 http://hdl.handle.net/123456789/2014071645714 eng Urheberrechtlich geschützt https://rightsstatements.org/page/InC/1.0/ Connection coefficients Linearization coefficients Duplications coefficients Classical orthogonal polynomials Askey-Wilson scheme 510 On Connection, Linearization and Duplication Coefficients of Classical Orthogonal Polynomials Dissertation In this work, we have mainly achieved the following: 1. we provide a review of the main methods used for the computation of the connection and linearization coefficients between orthogonal polynomials of a continuous variable, moreover using a new approach, the duplication problem of these polynomial families is solved; 2. we review the main methods used for the computation of the connection and linearization coefficients of orthogonal polynomials of a discrete variable, we solve the duplication and linearization problem of all orthogonal polynomials of a discrete variable; 3. we propose a method to generate the connection, linearization and duplication coefficients for q-orthogonal polynomials; 4. we propose a unified method to obtain these coefficients in a generic way for orthogonal polynomials on quadratic and q-quadratic lattices. Our algorithmic approach to compute linearization, connection and duplication coefficients is based on the one used by Koepf and Schmersau and on the NaViMa algorithm. Our main technique is to use explicit formulas for structural identities of classical orthogonal polynomial systems. We find our results by an application of computer algebra. The major algorithmic tools for our development are Zeilberger’s algorithm, q-Zeilberger’s algorithm, the Petkovšek-van-Hoeij algorithm, the q-Petkovšek-van-Hoeij algorithm, and Algorithm 2.2, p. 20 of Koepf's book "Hypergeometric Summation" and it q-analogue. open access Tcheutia, Daniel Duviol Universität, Kassel, Fachbereich Mathematik und Naturwissenschaften Koepf, Wolfram Foupouagnigni, Mama 33C45 33D45 Orthogonale Polynome 2014-07-14
The following license files are associated with this item:
:Urheberrechtlich geschützt