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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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Authors and Affiliations

  1. Dpto. de Matemática Aplicada, IUMA, Universidad de Zaragoza, Zaragoza, Spain

    Chelo Ferreira & Ester Pérez Sinusía

  2. Dpto. de Ingeniería Matemática e Informática, Universidad Pública de Navarra and INAMAT, Pamplona, Spain

    José L. López

Authors
  1. Chelo Ferreira
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  2. José L. López
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  3. Ester Pérez Sinusía
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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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