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Applications to Classes of Scalar Linear Difference Equations

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2129)

Abstract

In this chapter we are interested in scalar dth-order linear difference equations (also called linear recurrence relations) of the form \(\displaystyle{ y(n + d) = c_{1}(n)y(n) +\, \cdots +\, c_{d}(n)y(n + d - 1),\qquad n \in \mbox{ $\mathbb{N}$}, }\)

Keywords

  • Asymptotic Representation
  • Fundamental Matrix
  • Jacobi Operator
  • Exponential Dichotomy
  • Linear Difference Equation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. M.S.P. Eastham, The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem (Oxford University Press, Oxford, 1989)

    Google Scholar 

  2. S. Elaydi, An Introduction to Difference Equations (Springer, New York, 1999)

    CrossRef  MATH  Google Scholar 

  3. G. Freud, Orthogonal Polynomials (Pergamon Press, New York, 1971)

    Google Scholar 

  4. J. Geronimo, D.T. Smith, WKB (Liouville–Green) analysis of second order difference equations and applications. J. Approx. Theory 69, 269–301 (1992)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. J. Geronimo, D.T. Smith, Corrigendum to WKB (Liouville–Green) analysis of second order difference equations and applications. J. Approx. Theory 188, 69–70 (2014)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. J. Geronimo, W. Van Assche, Relative asymptotics for orthogonal polynomials with unbounded recurrence coefficients. J. Approx. Theory 62, 47–69 (1990)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. J. Geronimo, D.T. Smith, W. Van Assche, Strong asymptotics for orthogonal polynomials with regularly and slowly varying recurrence coefficients. J. Approx. Theory 72, 141–158 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. G.K. Immink, Asymptotics of Analytic Difference Equations. Lecture Notes in Mathematics, vol. 1085 (Springer, Berlin, 1984)

    Google Scholar 

  9. J. Janas, M. Moszyński, Spectral properties of Jacobi matrices by asymptotic analysis. J. Approx. Theory 120, 309–336 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. J. Janas, M. Moszyński, New discrete Levinson type asymptotics of solutions of linear systems. J. Differ. Equ. Appl. 12, 133–163 (2006)

    CrossRef  MATH  Google Scholar 

  11. J. Janas, S. Naboko, Jacobi matrices with absolutely continuous spectrum. Proc. Am. Math. Soc. 127, 791–800 (1999)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. S. Khan, D.B. Pearson, Subordinacy and spectral theory for infinite matrices. Helv. Phys. Acta 65, 505–527 (1992)

    MathSciNet  Google Scholar 

  13. A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials. Commun. Math. Phys. 179, 377–400 (1996)

    CrossRef  MATH  Google Scholar 

  14. R.J. Kooman, An asymptotic formula for solutions of linear second-order difference equations with regularly behaving coefficients. J. Differ. Equ. Appl. 13, 1037–1049 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. A. Máté, P. Nevai, Factorization of second-order difference equations and its application to orthogonal polynomials, in Orthogonal Polynomials and Their Applications (Segovia, 1986), ed. by M. Alfaro et al. Lecture Notes in Mathematics, vol. 1329 (Springer, Berlin, 1988), pp. 158–177

    Google Scholar 

  16. A. Máté, P. Nevai, A generalization of Poincaré’s theorem for recurrence equations. J. Approx. Theory 63, 92–97 (1990)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. A. Máté, P. Nevai, V. Totik, Asymptotics for orthogonal polynomials defined by a recurrence relation. Constr. Approx. 1, 231–248 (1985)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. H. Meschkowski, Differenzengleichungen. Studia Mathematica, Bd. XIV (Vandenhoeck & Ruprecht, Göttingen, 1959)

    Google Scholar 

  19. L.M. Milne-Thomson, The Calculus of Finite Differences (Macmillan and Co. Ltd., London, 1951)

    Google Scholar 

  20. P.G. Nevai, Orthogonal Polynomials. Memoirs of the American Mathematical Society, vol. 18 (American Mathematical Society, Providence, 1979)

    Google Scholar 

  21. O. Perron, Über einen Satz des Herrn Poincaré. J. Reine Angew. Math. 136, 17–37 (1909)

    MATH  Google Scholar 

  22. O. Perron, Über die Poincarésche lineare Differenzengleichung. J. Reine Angew. Math. 137, 6–64 (1910)

    Google Scholar 

  23. O. Perron, Über Systeme von linearen Differenzengleichungen erster Ordnung. J. Reine Angew. Math. 147, 36–53 (1917)

    MATH  Google Scholar 

  24. H. Poincaré, Sur les equations linéaires aux différentielles ordinaires et aux différences finies. Am. J. Math 7(3), 203–258 (1885)

    CrossRef  MATH  Google Scholar 

  25. T. Solda, R. Spigler, M. Vianello, Asymptotic approximations for second-order linear difference equations in Banach spaces, II. J. Math. Anal. Appl. 340, 433–450 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  26. R. Spigler, M. Vianello, Liouville–Green approximations for a class of linear oscillatory difference equations of the second order. J. Comput. Appl. Math. 41, 105–116 (1992)

    CrossRef  MATH  MathSciNet  Google Scholar 

  27. R. Spigler, M. Vianello, WKBJ-type approximation for finite moments perturbations of the differential equation y ′ ′ = 0 and the analogous difference equation. J. Math. Anal. Appl. 169, 437–452 (1992)

    Google Scholar 

  28. R. Spigler, M. Vianello, F. Locatelli, Liouville–Green–Olver approximations for complex difference equations. J. Approx. Theory 96, 301–322 (1999)

    CrossRef  MATH  MathSciNet  Google Scholar 

  29. S.A. Stepin, V.A. Titov, Dichotomy of WKB-solutions of discrete Schrödinger equation. J. Dyn. Control Syst. 12, 135–144 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  30. E.B. Van Vleck, On the extension of a theorem of Poincaré for difference-equations. Trans. Am. Math. Soc. 13, 342–352 (1912)

    CrossRef  MATH  Google Scholar 

  31. J. Wimp, D. Zeilberger, Resurrecting the asymptotics of linear recurrences. J. Math. Anal. Appl. 111, 162–176 (1985)

    CrossRef  MATH  MathSciNet  Google Scholar 

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Bodine, S., Lutz, D.A. (2015). Applications to Classes of Scalar Linear Difference Equations. In: Asymptotic Integration of Differential and Difference Equations. Lecture Notes in Mathematics, vol 2129. Springer, Cham. https://doi.org/10.1007/978-3-319-18248-3_9

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