tailieunhanh - báo cáo hóa học:" Research Article Transformations of Difference Equations I"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Transformations of Difference Equations I | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 947058 22 pages doi 2010 947058 Research Article Transformations of Difference Equations I Sonja Currie and Anne D. Love School of Mathematics University of the Witwatersrand Private Bag 3 PO WITS 2050 Johannesburg South Africa Correspondence should be addressed to Sonja Currie Received 13 April 2010 Revised 27 July 2010 Accepted 29 July 2010 Academic Editor Mariella Cecchi Copyright 2010 S. Currie and A. D. Love. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We consider a general weighted second-order difference equation. Two transformations are studied which transform the given equation into another weighted second order difference equation of the same type these are based on the Crum transformation. We also show how Dirichlet and non-Dirichlet boundary conditions transform as well as how the spectra and norming constants are affected. 1. Introduction Our interest in this topic arose from the work done on transformations and factorisations of continuous as opposed to discrete Sturm-Liouville boundary value problems by amongst others Binding et al. notably 1 2 . We make use of similar ideas to those discussed in 3-5 to study the transformations of difference equations. In this paper we consider a weighted second-order difference equation of the form ly -c n y n 1 b n y n - c n - 1 y n - 1 c ri fy ri where cfri 0 represents a weight function and bfn a potential function. Two factorisations of the formal difference operator l associated with are given. Although there may be many alternative factorisations of this operator see . 2 6 the factorisations given in Theorems and are of particular interest to us as they are analogous to those used in the continuous .