Irrationality Measures for the Series of Reciprocals from Recurrence Sequences

Abstract

Using Pade´approximations of Heine's q-hypergeometric series we obtain new irrationality measures for the values of the series P 1 n¼1 t n =W n ; where W n is a Fibonacci or Lucas type arithmetical form satisfying the recurrence

Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers

“…Here we may suppose without a loss of generality that |α| > |β| and hence F n L n = 0 for all n ∈ Z + . André-Jeannin [1] proved the irrationality of the series (2) in the Fibonacci case, where a = b = 1, and soon after followed some irrationality measure considerations of the series (2) in the case a = 1, see [4,9,12,13]. However, not much is known about the arithmetic character of the series (2) with arbitrary parameters a, b except the transcendence coming from Nesterenko's method [6,10] for the numbers…”

Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers

“…More generally, P. Bundschuh and K. Väänänen [2] obtained f / ∈ Q( √ 5) as well as an irrationality measure. Much is known about the quantitative result of f ; see, e.g., [9,10,11] on this direction. On the other hand, we know very little about linear independence results; for example, of the three numbers 1, f ,…”