an:03081783
Zbl 0051.27801
Lech, Christer
A note on recurring series
EN
Ark. Mat. 2, 417-421 (1953).
00130328
1953
j
11B37
The author generalizes a result by \textit{Th. Skolem} [8. Skand. Mat.-Kongr., 163--188 (1935; Zbl 0011.39201)] and \textit{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.
K. Mahler
Zbl 0011.39201; Zbl 0010.39005