The method of variation of parameters in the theory of linear sequences. (English) Zbl 0755.39001

Consider a linear differential equation of the form \[ f[\phi_ p(x)]- p_ 1(x)f(\phi_{p-1}(x))-\dots-p_ p(x)f(\phi_ 0(x))=0\qquad x\in(- \infty,\infty) \] which is defined over a cyclic group of functions \(G=\{\phi_ r(t)\}^ \infty_{r=-\infty}\). The author shows that the solutions of this equation on a set of points \(\{t_ r\}^ \infty_{r=0}\), where \(t_ r=\phi_ r(t_ 0)\), \(t_ 0\in(- \infty,\infty)\) are just only general linear sequences defined by \(x_ 1=a_ 1,\dots,x_ p=a_ p\), \(x_ n=\alpha_{n-p}x_{n- 1}+\dots+\alpha_{pn-p}x_{n-p}\) for \(n=p+1,p+2,\dots\). The method of variation of stationary sequences is modified to these sequences.


39A10 Additive difference equations
Full Text: EuDML


[1] Gelfond A.O.: Diferenzenrechnung. VEB VdW Berlin, 1958.
[2] Laitoch M.: Lineární posloupnosti. Acta Universitatis Palackianae Olomoucensis, FRN 2 (1968), 51-67.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.