Abstract
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.
Highlights
A linear difference equation with nonconstant coefficients has the following general form: aN (n) fn+N + aN−1(n) fn+N−1 + · · · + a1(n) fn+1 + a0(n) fn = gn, (1.1)where the functions a0, a1, . . . , aN, g : N → C are given with aN not identically zero, and N, k are fixed nonnegative integers
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
Linear difference equations with nonconstant coefficients present another important class of difference equations with much less highly developed methods and theories
Summary
A linear difference equation with nonconstant coefficients has the following general form: aN (n) fn+N + aN−1(n) fn+N−1 + · · · + a1(n) fn+1 + a0(n) fn = gn,. 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, . In this case the solution space is generated by exponential monomial solutions, which arise from the roots of the characteristic polynomial, called characteristic roots.
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