On Diophantine approximations of Ramanujan type q-continued fractions

Abstract

We shall consider arithmetical properties of the q-continued fractionsand some related continued fractions where v is a fixed valuation of an algebraic number field K and s, h, l ∈ N. In particular, we get sharp irrationality measures for certain Ramanujan, RamanujanSelberg, Eisenstein and Tasoev continued fractions.

On irrationality exponents of generalized continued fractions

“…In particular, when a = 1 we get µ I (RR( 1 b , t)) = 2 for all b ∈ Z + , b ≥ 2, a result proved already by Bundschuh [4]. For more general approximation measures for q-continued fractions, see [14]. In [14] we may find an example M(q) = ∞ K n=1 q 2n 1 + q n where q = a b , a, b ∈ Z + .…”

“…For more general approximation measures for q-continued fractions, see [14]. In [14] we may find an example M(q) = ∞ K n=1 q 2n 1 + q n where q = a b , a, b ∈ Z + . Similarily to the previous example we get µ I (M(q)) ≤ 2 + 2 log a log b − 2 log a when a 2 < b.…”

“…Next we shall consider Tasoev's continued fractions previously studied e.g. in [9], [10] and [14]. We define and T 2 that we can apply Theorem 5.1.…”

Irrationality Measures of Numbers Related to Some $q$-Basic Hypergeometric Series

“…We note that there are a lot of works considering arithmetic properties of different type of q-series, see e.g. [14] for a survey of such results, and [3], [4], [5], [6], [8], [10], [11], [12] and [15] for some more recent results, but only a few study functions (2), see [7] and [10]. In [7] Chirskii considered the case K = Q with finite p and proved the following result.…”

“…(1 − a)f (aq, b, z) = f (a, b, z) − af (a, b, qz), Theorem 1.1 gives also linear independence of p-adic numbers f (ξ) and f (qξ). The above theorem was generalized and made quantitative in [10], where the authors study the solutions of functional equations…”