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MATHEMATICA BOHEMICA, Vol. 128, No. 3, pp. 293-308 (2003)
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Phases of linear difference equations and symplectic systems

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Zuzana Dosla, Denisa Skrabakova

* Zuzana Dosla*, * Denisa Skrabakova*, Department of Mathematics, Masaryk University, Janackovo nam. 2a, 662 95 Brno, Czech Republic, e-mails: ` dosla@math.muni.cz`, ` denisa@math.muni.cz`

**Abstract:** The second order linear difference equation

\Delta(r_k\triangle x_k)+c_kx_{k+1}=0,\tag1

where $r_k\ne0$ and $k\in\mathbb{Z}$, is considered as a special type of symplectic systems. The concept of the phase for symplectic systems is introduced as the discrete analogy of the Boruvka concept of the phase for second order linear differential equations. Oscillation and nonoscillation of (1) and of symplectic systems are investigated in connection with phases and trigonometric systems. Some applications to summation of number series are given, too.

**Keywords:** second order linear difference equation, symplectic system, phase, oscillation, nonoscillation, trigonometric transformation

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