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second order linear difference equation; symplectic system; phase; oscillation; nonoscillation; trigonometric transformation
The second order linear difference equation \[ \Delta (r_k\triangle x_k)+c_kx_{k+1}=0, \qquad \mathrm{(1)}\] where $r_k\ne 0$ 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 Borůvka 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.
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