Asymptotic behavior of solutions of three-term Poincaré difference equations. (English) Zbl 0888.39006
The author considers the difference equation \(Z_n= b_nZ_{n-1} +a_n Z_{n-2}\) with complex coefficients \(a_n\to a\), \(b_n\to b\) for \(n\to\infty\) and \(a_n\neq 0\). The solutions \(x_1\), \(x_2\) of \(x^2= ax+b\) shall satisfy \(|x_1|<|x_2|\). Under the assumption that \(\sum(a_n-a)\) and \(\sum(b_n-b)\) are only conditionally convergent, conditions are given such that \(\lim_{n\to\infty} Z_nx_2^{-n}=Z\) exists, and the error \(|Z-Z_nx_2^{-n} |\) is estimated. The results are applied to ten examples, in particular to recurrence equations arising from hypergeometric functions and continued fractions, respectively.
Reviewer: Lothar Berg (Rostock)
MSC:
| 39A11 | Stability of difference equations (MSC2000) |
| 11B37 | Recurrences |
| 40A15 | Convergence and divergence of continued fractions |
Keywords:
asymptotic behavior; three-term Poincaré difference equations; recurrence equations; hypergeometric functions; continued fractionsReferences:
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