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: DIFFERENCE EQUATIONS ON DISCRETE POLYNOMIAL HYPERGROUPS | DIFFERENCE EQUATIONS ON DISCRETE POLYNOMIAL HYPERGROUPS AGOTA OROSZ Received 10 July 2005 Revised 24 October 2005 Accepted 30 October 2005 The classical theory of homogeneous and inhomogeneous linear difference equations with constant coefficients on the set of integers or nonnegative integers provides effective solution methods for a wide class of problems arising from different fields of applications. However linear difference equations with nonconstant coefficients present another important class of difference equations with much less highly developed methods and theories. In this work we present a new approach to this theory via polynomial hypergroups. It turns out that a major part of the classical theory can be converted into hypergroup language and technique providing effective solution methods for a wide class of linear difference equations with nonconstant coefficients. Copyright 2006 Agota Orosz. 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. 1. Introduction A linear difference equation with nonconstant coefficients has the following general form aN n fn N aN-1 n fn N-1 ffi n fn 1 o n fn gn where the functions a0 a1 . aN g N C are given with aN not identically zero and N k are fixed nonnegative integers in this paper N 0 1 2 . . The above equation is supposed to hold for some unknown function f N C or f Z C depending on the nature of the problem. In what follows we will prefer the case f N C and the notation f m and g m instead of fm and gm. By the classical theory of differential equations the solution space of the above equation can be described completely in the constant coefficient case that is if the functions a0 a1 . aN are constants. In this case the solution space is generated by exponential monomial solutions which arise from the roots of the characteristic polynomial called .
