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

  • Chelo Ferreira
  • José L. LópezEmail author
  • Ester Pérez Sinusía


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.


Second-order difference equations asymptotic expansions Green’s functions Olver’s method 

Mathematics Subject Classification

39A06 41A58 41A60 34B27 


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Authors and Affiliations

  1. 1.Dpto. de Matemática Aplicada, IUMAUniversidad de ZaragozaZaragozaSpain
  2. 2.Dpto. de Ingeniería Matemática e InformáticaUniversidad Pública de Navarra and INAMATPamplonaSpain

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