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Convergent and Asymptotic Methods for Second-order Difference Equations with a Large Parameter

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Abstract

We consider the second-order linear difference equation \(y(n+2)-2a y(n+1)-\Lambda ^2 y(n)=g(n)y(n)+f(n)y(n+1)\), where \(\Lambda \) is a large complex parameter, \(a\ge 0\) and g and f are sequences of complex numbers. Two methods are proposed to find the asymptotic behavior for large \(\vert \Lambda \vert \) of the solutions of this equation: (i) an iterative method based on a fixed point method and (ii) a discrete version of Olver’s method for second-order linear differential equations. Both methods provide an asymptotic expansion of every solution of this equation. The expansion given by the first method is also convergent and may be applied to nonlinear problems. Bounds for the remainders are also given. We illustrate the accuracy of both methods for the modified Bessel functions and the associated Legendre functions of the first kind.

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References

  1. Ahlbrandt, C.D., Peterson, A.C.: Discrete Hamiltonian Systems. Springer-Science+Busnisess Media, B.V., New York (1996)

    Book  Google Scholar 

  2. Alsharawi, Z., Cushing, J.M., Elaydi, S. Eds., Theory and Applications of Difference Equations and Discrete Dynamical Systems. In: Springer Proceedings in Mathematics and Statistics, Springer, New York (2013)

  3. Cao, L., Li, Y.: Linear difference equations with a transition point at the origin. Anal. Appl. (Singap.) 12(1), 75–106 (2014)

    Article  MathSciNet  Google Scholar 

  4. Cash, J.R.: An extension of Olver’s method for the numerical solution of linear recurrence relations. Math. Comput. 32(2), 497–510 (1978)

    MathSciNet  MATH  Google Scholar 

  5. Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York (1960)

    MATH  Google Scholar 

  6. Dunster, T.M.: Legendre and Related Functions, in NIST Handbook of Mathematical Functions, U.S. Department of Commerce, Washington, DC (2010). http://dlmf.nist.gov/14

  7. Elaydi, S.: An Introduction to Difference Equations. Springer, New York (2005)

    MATH  Google Scholar 

  8. Ferreira, C., López, J.L., Pérez Sinusía, E.: Convergent and asymptotic expansions of solutions of differential equations with a large parameter: Olver cases II and III. J. Integral Equ. Appl. 27(1), 27–45 (2015)

    Article  MathSciNet  Google Scholar 

  9. Ferreira, C., López, J.L., Pérez Sinusía, E.: Convergent and asymptotic expansions of solutions of second-order differential equations with a large parameter. Anal. Appl. (Singap.) 12(5), 523–536 (2014)

    Article  MathSciNet  Google Scholar 

  10. Fulford, G., Forrester, P., Jones, A.: Modeling with Differential and Difference Equations. Cambridge University Press, New York (1997)

    Book  Google Scholar 

  11. Geronimo, J.S., Smith, D.T.: WKB (Liouville-Green) analysis of second order difference equations and applications. J. Approx. Theory 69(3), 269–301 (1992)

    Article  MathSciNet  Google Scholar 

  12. Gil, A., Segura, J., Temme, N.M.: Numerical Methods for Special Functions. SIAM, Philadelphia (2007)

    Book  Google Scholar 

  13. Goldberg, S.: Introduction to Difference Equations. Dover, New York (1986)

    Google Scholar 

  14. Jovanović, B.S.: Analysis of Finite Difference Schemes. Springer, London (2014)

    Book  Google Scholar 

  15. Kelley, W.G., Peterson, A.C.: Difference Equations. An Introduction with Applications. Academic Press, New York (1991)

    MATH  Google Scholar 

  16. Koornwinder, T.H., Wong, R., Koekoek, R., Swarttouw, R.F.: Orthogonal Polynomials. In: NIST Handbook of Mathematical Functions, U.S. Department of Commerce, Washington, DC (2010). http://dlmf.nist.gov/10

  17. Levy, A.: Economic Dynamics. Avebury, Hong Kong (1992)

    Google Scholar 

  18. López, J.L., Temme, N.: Approximations of orthogonal polynomials in term of Hermite polynomials. Meth. Appl. Anal. 6(2), 131–146 (1999)

    MathSciNet  MATH  Google Scholar 

  19. López, J.L.: Olver’s asymptotic method revisited. Case I. J. Math. Anal. Appl. 395(2), 578–586 (2012)

    Article  MathSciNet  Google Scholar 

  20. Mickens, R.E.: Difference Equations. Theory, Applications and Advanced Topics. CRC Press, New York (2015)

    MATH  Google Scholar 

  21. Olver, F.W.J.: Bounds for the solutions of second-order linear difference equations, TJ. Res. Nat. Bur. Stand. Sect. B 71B, 111–129 (1967)

    Article  MathSciNet  Google Scholar 

  22. Olver, F.W.J.: Numerical solution of second-order linear difference equations. TJ. Res. Nat. Bur. Stand. Sect. B 71B, 161–166 (1967)

    Article  MathSciNet  Google Scholar 

  23. Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)

    MATH  Google Scholar 

  24. Olver, F.W.J., Maximon, L.C.: Bessel Functions, in NIST Handbook of Mathematical Functions, U.S. Department of Commerce, Washington, DC, (2010). http://dlmf.nist.gov/10

  25. Sedaghat, H.: Nonlinear Difference Equations. Theory with Applications to Social Science Models. Springer-Science+Busnisess Media, B.V, New York (2003)

    MATH  Google Scholar 

  26. Sharkovsky, A.N., Maistrenko, Y.L., Yu Romanenko, E.: Difference Equations and their Applications. Springer, Dordrecht (1993)

    Book  Google Scholar 

  27. Schwab, A.J.: Field Theory Concepts. Springer, New York (1988)

    Book  Google Scholar 

  28. Spigler, R., Vianello, M.: Liouville-Green approximations for a class of linear oscillatory difference equations of the second order. J. Comput. Appl. Math. 41(1–2), 105–116 (1992)

    Article  MathSciNet  Google Scholar 

  29. Spigler, R., Vianello, M., Locatelli, F.: Liouville-Green-Olver approximations for complex difference equations. J. Approx. Theory 96(2), 301–322 (1999)

    Article  MathSciNet  Google Scholar 

  30. Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge University Press, New York (2012)

    MATH  Google Scholar 

  31. Wong, R.: Asymptotics of linear recurrences. Anal. Appl. (Singap.) 12(4), 463–484 (2014)

    Article  MathSciNet  Google Scholar 

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Correspondence to José L. López.

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Ferreira, C., López, J.L. & Pérez Sinusía, E. Convergent and Asymptotic Methods for Second-order Difference Equations with a Large Parameter. Mediterr. J. Math. 15, 224 (2018). https://doi.org/10.1007/s00009-018-1267-9

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  • DOI: https://doi.org/10.1007/s00009-018-1267-9

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