Journal of Nonlinear Mathematical Physics

Volume 15, Issue supplement 3, October 2008, Pages 73 - 80

Nonlinear-Integral-Equation Construction of Orthogonal Polynomials

Authors
Carl M. Bender, E. Ben-Naim
Corresponding Author
Carl M. Bender
Available Online 1 October 2008.
DOI
10.2991/jnmp.2008.15.s3.8How to use a DOI?
Abstract

The nonlinear integral equation P(x) = dyw(y)P(y)P(x + y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions Pn(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials. Interestingly, the set of polynomial solutions is orthogonal with respect to the measure xw(x). The nonlinear integral equation can be used to specify all orthogonal polynomials in a simple and compact way. This integral equation provides a natural vehicle for extending the theory of orthogonal polynomials into the complex domain. Generalizations of the integral equation are discussed. Finally, it is observed that since the integral equation is independent of the degree of the polynomials it may possibly be a useful tool in determining and studying the asymptotic behaviors of polynomials.

Copyright
© 2008, the Authors. Published by Atlantis Press.
Open Access
This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
15 - supplement 3
Pages
73 - 80
Publication Date
2008/10/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.2991/jnmp.2008.15.s3.8How to use a DOI?
Copyright
© 2008, the Authors. Published by Atlantis Press.
Open Access
This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Carl M. Bender
AU  - E. Ben-Naim
PY  - 2008
DA  - 2008/10/01
TI  - Nonlinear-Integral-Equation Construction of Orthogonal Polynomials
JO  - Journal of Nonlinear Mathematical Physics
SP  - 73
EP  - 80
VL  - 15
IS  - supplement 3
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2008.15.s3.8
DO  - 10.2991/jnmp.2008.15.s3.8
ID  - Bender2008
ER  -