Abstract
Liouville–Green (WKB) asymptotic approximations are constructed for some classes of linear second-order difference equations. This is done starting from certain “canonical forms” for the three-term linear recurrence. Rigorous explicit bounds are established for the error terms in the asymptotic approximations of recessive as well as dominant solutions. The asymptotics with respect to parameters affecting the equation is also discussed. Several illustrative examples are given.
Highlights
In this paper we first derive certain “canonical forms” for the three-term linear recurrence, and we use them to obtain asymptotic approximations of the Liouville–Green (LG, for short, or WKB) type for their solutions
3 The main results we develop an LG asymptotic approximation theory for the three classes of linear second-order difference equations (B), (C), and (D), along the same lines followed for the form (A) in [14]
Starting from the canonical forms to establish LG asymptotic approximations is convenient, since we are able to exploit their similarity with second-order differential equations with no first-order derivative terms (Jacobi or Sturm–Liouville forms)
Summary
In this paper we first derive certain “canonical forms” for the three-term linear recurrence, and we use them to obtain asymptotic approximations of the Liouville–Green (LG, for short, or WKB) type for their solutions. Starting from the canonical forms to establish LG asymptotic approximations is convenient, since we are able to exploit their similarity with second-order differential equations with no first-order derivative terms (Jacobi or Sturm–Liouville forms). This approach parallels closely the original one followed by F.W.J. Olver for differential equations [9]. Equation (36) can be written as [ εn + βεn] = 0, which suggests that two linearly independent solutions can be promptly obtained, one given by εn ≡ 1, and the other constructed solving εn + βεn = 0, which yields εn = ηn,.
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