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Results on the ratios of the terms of second order linear recurrences. (English) Zbl 0758.11008

Several results on the diophantine approximation by second order linear recurrences are surveyed. One such result is as follows: Let \(R\) be a nondegenerate second order linear recurrence with negative discriminant. Then there is a constant \(c>0\) such that (for the characteristic root \(\alpha\)):
\[ \bigl| |\alpha| - | R_{n+1}/R_ n| \bigr| <1/n^ c \]
for infinitely many \(n\). This bound is best possible apart from the value of the constant \(c\), i.e. for sufficiently large \(n\) the converse inequality is true with another constant [cf. P. Kiss and R. F. Tichy, Proc. Japan Acad., Ser. A 65, 135-138, 191-194 (1989; Zbl 0692.10041)].
Reviewer: R.F.Tichy (Graz)

MSC:

11B37 Recurrences
11J04 Homogeneous approximation to one number

Citations:

Zbl 0692.10041
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References:

[1] JARDEN D.: Recurring Sequences. Riveon Lematematika, Jerusalem (Israel), 1973. · Zbl 0035.30802
[2] KISS P.: Zero terms in second order linear recurrences. Math. Sem. Notes (Kobe Univ.) 7 (1979), 145-152. · Zbl 0417.10009
[3] KISS P.: A Diophantine approximative property of the second order linear recurrences Period. Math. Hungar. 11 (1980), 281-287. · Zbl 0458.10011
[4] KISS P.: On second order recurrences and continued fractions. Bull. Malaysian Math Soc. (2) 5 (1982), 33-41. · Zbl 0499.10010
[5] KISS P., SINK A Z.: On the ratios of the terms of second order linear recurrences.
[6] KISS P., TICHY R. F.: A discrepancy problem with applications to linear recurrences I, II. Proc. Japan Acad. 65, Ser. A (1989), 135-138, 191-194. · Zbl 0692.10041
[7] MÁTYÁS F.: Másodrendü lineáris rekurziv sorozatok elemeinek hányadosairól. (Hungarian), Math. Lapok 27 (1976-1979), 379-389.
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