Irrationality of Growth Constants Associated with Polynomial Recursions
Stephan Wagner
Department of Mathematics
Uppsala University
Box 480
751 06 Uppsala
Sweden
and
Department of Mathematical Sciences
Stellenbosch University
Private Bag X1
Matieland 7602
South Africa
Volker Ziegler
Department of Mathematics
University of Salzburg
Hellbrunnerstrasse 34/I
5020 Salzburg
Austria
Abstract:
We consider integer sequences that satisfy a recursion of the form
xn+1 = P(xn)
for some polynomial P of degree d > 1. If such a
sequence tends to infinity, then it satisfies an asymptotic formula of the
form xn ∼ A αdn, but little can be said about the constant
α. In this paper, we show that α is always irrational or
an integer. In fact, we prove a stronger statement: if a sequence
(Gn)n ≥ 0 satisfies an
asymptotic formula of the form Gn = A
αn + B + O(α-ε
n), where A, B are algebraic and α > 1,
and the sequence contains infinitely many integers, then α is
irrational or an integer.
Full version: pdf,
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(Concerned with sequences
A000058
A003095
A076949.)
Received August 12 2020;
revised versions received August 13 2020; December 31 2020; January 1 2021.
Published in Journal of Integer Sequences,
January 2 2021.
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