## 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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