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Asymptotic behavior of solutions of general three term recurrence relations. (English) Zbl 1160.39001

The authors consider the following recurrence relation
\[ w_n = b_n(z)w_{n-1} - a^2_n(z)w_{n-2},\quad n \in {\mathbb N},\;z \in \Omega, \]
where \(\Omega \subseteq {\mathbb C}\), \(a_n(z) \neq 0\) and \(b_n(z)\) are analytic functions. Let \(N \in {\mathbb N}\) be fixed. It is assumed that the locally uniform convergences hold
\[ \lim_{m \to \infty} a_{mN+j}(z) = a^{(j)}(z),\quad \lim_{m \to \infty} b_{mN+j}(z) = b^{(j)}(z),\;j = 0, 1,\dots, N-1,\;z \in \Omega, \]
\[ \text{and }a^{(j)}(z) \neq 0,\quad j = 0, 1,\dots, N-1,\;z \in \Omega. \]
The asymptotic behavior of the solutions is studied.

MSC:

39A10 Additive difference equations
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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References:

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