Orthogonal and similar polynomials: properties and applications

• Grant number: 99/12054-9
• Support Opportunities: Research Projects - Thematic Grants
• Start date: March 01, 2000
• End date: February 28, 2003
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Alagacone Sri Ranga
• Grantee: Alagacone Sri Ranga
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Associated research grant(s): 02/05311-0 - Xin Li | University of Central Florida - Estados Unidos, AV.EXT

Abstract

Our involvement in the studies of orthogonal and similar polynomials has been a fairly successful undertaking for many years. This area of research, which has attracted the interest of many famous mathematicians, has become an important and extremely active area of study in recent years. We hope that our participation in this area of study will be even more intensive and that the benefits that come with this participation are, apart from being able to get new results that lead to new research articles, that we can also pass the newly acquired knowledge to our students, who are part of the future generation of Brazilian mathematicians. With this objective, we will not only carry out the topics of research mentioned below, we will also continuously look for other problems and ideas that will permit our group to remain active and productive.to consider an extension of the studies involving positive chain sequences to generalized chain sequences {(1- hn-1)hn= an}8n=0, by removing the condition that hn must satisfy 01. The use of these chain sequences, together with results that we already have, provides other means of studying certain orthogonal polynomials frorn the behaviour of some wellknown orthogonal polynomials. In particular, as a first step, wc will study the Koornwinder [N15] and Krall [N16] polynomials using the behaviour of the Gegenbaur polynomials. The idea of exploring the symmetries in the measures to study the consequent properties andapplications of orthogonal L-polynomials was originally due to our group. In a number of articles, we have proved many results concerning the polynomials Bn(y1,..., yr; w; t) =Bwn (t) + E rj=1 yjBwn=j(t) by considering the classes of symmetries denoted by S3 (w, ß, b). Recently, our efforts have been focused on a detailed study of the polynomials Bn(y1, y2; w; t) when the measure dw is of the class S3 (-1/2, ß, b). We have already obtained some interesting properties of these polynomials by considering different choice of the parameters y1 and y2. Still there are many problems and questions that need to be answered. Following the order in which we have been exploring the symmetries, the next stage would be to consider a detailed study of the polynomials Bn(y1, y2, y3; w ; t) when dw E =S3 (-1, ß, b). However, with the increase in the number of parameters y, the mathematical analysis involved is more and more complex, requiring also a lot of computational aid to reach our objectives. To explore the relations that exists between the well known Szegõ polynomials Sn(z) and certain orthogonal L-polynomials. (AU)