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A note on recurring series. (English) Zbl 0051.27801
The author generalizes a result by Th. Skolem [8. Skand. Mat.-Kongr., 163–188 (1935; Zbl 0011.39201)] and K. Mahler [Mathematica, Leiden 3, 153–156 (1934; Zbl 0010.39005)], and proves the following theorem:
In a field of characteristic 0, let \(c_\nu\) \(\nu=0,1,2,\dots)\) satisfy the recursive formula \[ c_\nu=\alpha_1c_{\nu-1}+\alpha_2c_{\nu-2}+\dots+\alpha_nc_{\nu-n}\quad (\nu=n+1,n+2,\dots). \]
If \(c_\nu=0\) for infinitely many \(\nu\), then these vanishing \(c_\nu\) occur periodically in the sequence from a certain \(\nu\) onwards.
The proof is similar to that of Mahler. The main difficulty arises in the case when the field generated by all \(c_\nu\) is transcendental over the rational field.
Reviewer: K. Mahler

MSC:
11B37 Recurrences
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References:
[1] K. Hensel, Theorie der algebraischen Zahlen I, Leipzig und Berlin, 1908.
[2] K. Mahler, Eine arithmetische Eigenschaft der Taylor-koeffizienten rationaler Funktionen, Akad. Wetensch. Amsterdam, Proc. 38, 50–60 (1935). · JFM 61.0176.02
[3] Th. Skolem, Einige Sätze über gewisse Reihenentwicklungen und exponentiale Beziehungen mit Anwendung auf diophantische Gleichungen. Oslo Vid. akad. Skrifter I 1933 Nr. 6.
[4] Th. Skolem, Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen, C. r. 8 congr. scand. à Stockholm 1934, 163–188. · JFM 61.1080.01
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