Floquet theory for second order linear homogeneous difference equations |
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Authors: | A.M. Encinas M.J. Jiménez |
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Affiliation: | 1. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain.andres.marcos.encinas@upc.edu;3. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain. |
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Abstract: | In this paper we provide a version of the Floquet’s theorem to be applied to any second order difference equations with quasi-periodic coefficients. To do this we extend to second order linear difference equations with quasi-periodic coefficients, the known equivalence between the Chebyshev equations and the second order linear difference equations with constant coefficients. So, any second order linear difference equations with quasi-periodic coefficients is essentially equivalent to a Chebyshev equation, whose parameter only depends on the values of the quasi-periodic coefficients and can be determined by a non-linear recurrence. Moreover, we solve this recurrence and obtaining a closed expression for this parameter. As a by-product we also obtain a Floquet’s type result; that is, the necessary and sufficient condition for the equation has quasi-periodic solutions. |
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Keywords: | Difference equations Floquet theory periodic sequences Chebyshev polynomials |
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