Algebraic independence results for reciprocal sums formed by

Algebraic independence results for reciprocal sums formed by

Institut f. Analysis und Zahlentheorie
Zahlentheoretisches Kolloquium
Freitag, 11. 3. 2016, 14:00 Uhr
SR NT02008, Kopernikusgasse 24/2.Stock
Algebraic independence results for reciprocal sums formed by
powers of Fibonacci numbers
Prof. Dr. Carsten Elsner
(Fachhochschule für die Wirtschaft Hannover)
Abstract:
Let (Un )n≥0 be a sequence of numbers given by Un = (αn − β n )/(α − β) for n ≥ 0,
where α and β are complex numbers satisfying |β| < 1 and αβ = −1. Let s ≥ 1
and
∞
X
1
−2s
Φ2s := (α − β)
.
U 2s
n=1 n
In a joint paper of the lecturer with Prof. Shun Shimomura and Prof. Iekata Shiokawa it is shown that for any positive integers s1 , s2 , s3 corresponding to algebraic
α, β with |β| < 1 and αβ = −1 the numbers Φ2s1 , Φ2s2 , Φ2s3 are algebraically√independent over √
if and only if s1 s2 s3 ≡ 0 (mod 2). In particular, for β = (1 − 5)/2
and β = 1 − 2 the results on Φ2s are applicable to reciprocal sums formed by
powers of Fibonacci numbers Un = Fn and by powers of Pell numbers Un = Pn , respectively. In the lecture the proof of the algebraic independence of Φ2s1 , Φ2s2 , Φ2s3
is presented for the particular case of three even integers 2 ≤ s1 < s2 < s3 .
N. Technau, R. Tichy