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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.

MSC:

39A10 Additive difference equations
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References:

[1] Gelfond A.O.: Diferenzenrechnung. VEB VdW Berlin, 1958.
[2] Laitoch M.: Lineární posloupnosti. Acta Universitatis Palackianae Olomoucensis, FRN 2 (1968), 51-67.
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