ITEM METADATA RECORD
Title: Multiple Charlier and Meixner polynomials
Other Titles: Meervoudige Charlier en Meixner veeltermen
Authors: Ndayiragije, François; S0208557
Issue Date: 10-Jul-2012
Abstract: We investigate multiple Charlier and multiple Meixner polynomials. These are extensions of the classical Charlier and Meixner polynomials which are orthogonal polynomials with respect to the Poisson distribution and the Pascal distribution (negative binomial distribution). For multiple C harlier polynomials we impose orthogonality conditions with respect to&n bsp;r different Poisson distributions (with r > 1) and for multiple Meixner polynomials we use r different Pasc al distributions. Depending on the choice of which parameters we select, we then get multiple Meixner polynomials of the first or of the second kind. We give a generating function for multiple Charlier polynomials an d we explicitly compute the recurrence coefficients for the recurrence r elations connecting the nearest neighbor multiple Charlier polynomials. The most important result is the asymptotic behavior of the ratio of two neighboring multiple Charlier polynomials and the asymptotic behavior o f the distribution of the (scaled) zeros, which turn out to be uniformly distributed, independent of the choice of the parameters of the Poisson distributions. We also investigate the case where one of the parameters depends on the scaling and then find a combination of a uniform distrib ution and a new distribution, which in some cases lives on an interval d isjoint of the interval where the scaled zeros are uniformly distributed . We also give a generating function for multiple Meixner polynomials of the first and of the second kind and we compute the coefficients in the nearest neighbor recurrence relations explicitly. Using these formulas, we are able to build on recent results of Miki, Tsujimoto, Vinet and Zh edanov who introduced non-Hermitian oscillator Hamiltonians for which th e eigenstates can be expressed in terms of multiple Charlier polynomials or multiple Meixner polynomials of the first kind. We extend this by co nstructing non-Hermitian oscillator Hamiltonians for which the eigenstat es can be described in terms of multiple Meixner polynomials of the seco nd kind (for r = 2).
Table of Contents: Summary

1. Introduction
1.1 Orthogonal polynomials on the real line
1.2 The classical orthogonal polynomials on the real line
1.3 Askey scheme of hypergeometric orthogonal polynomials
1.4 Multiple orthogonal polynomials
1.5 Outline of the thesis

2. Multiple Charlier polynomials
2.1 Introduction
2.2 Some properties of multiple Charlier polynomials
2.3 Ratio asymptotics
2.4 Asymptotic distribution of the zeros
2.5 Parameters depending on the degree
2.6 Concluding remarks

3. Multiple Meixner polynomials
3.1 Introduction
3.2 Generating function
3.3 Recurrence relations
3.4 Concluding remarks

4. Some applications of multiple orthogonal polynomials: Non-Hermitian oscillator Hamiltonians
4.1 One-dimensional harmonic oscillator
4.2 r-dimensional harmonic oscillator
4.3 Multiple Charlier polynomials and non-Hermitian oscillator Hamiltonians
4.4 Multiple Meixner polynomials of the first kind and non-Hermitian oscillator Hamiltonians
4.5 Multiple Meixner polynomials of the second kind and non-Hermitian oscillator Hamiltonians

5. Conclusion and outlook for future research

Nederlandse samenvatting

Bibliography
ISBN: 978-90-8649-543-6
Publication status: published
KU Leuven publication type: TH
Appears in Collections:Analysis Section

Files in This Item:
File Status SizeFormat
Francoisphd-final.pdf Published 541KbAdobe PDFView/Open

These files are only available to some KU Leuven staff members

 


All items in Lirias are protected by copyright, with all rights reserved.