Non-converging continued fractions related to the Stern diatomic sequence

Abstract

Abstract. -This note is essentially an addendum to the recent article of Dilcher and Stolarsky [7] though some results presented here may be of independent interest. We prove the transcendence of some irregular continued fractions which are related to the Stern diatomic sequence. The proofs of our results rest on the so-called Mahler method.

Transcendence tests for Mahler functions

“…As neither of these roots is a power of 2, both F (z) and G(z) are transcendental. This result was previous proved by us [6] and independently by Adamczewski [1].…”

“…A function F (z) ∈ C[z] is called a k-Mahler function provided there are integers k 2 and d 1 such that (1) a 0 (z)F (z) + a 1 (z)F (z k ) + · · · + a d (z)F (z k d ) = 0, for some polynomials a 0 (z), . .…”

Stern polynomials and double-limit continued fractions

“…The main theorem in [20] does not apply directly to (5.5) and (5.6), but using a result of Nishioka [21], Adamczewski [1] was able to prove Proposition 5.4 (Adamczewski). The functions F and G take transcendental values at every algebraic number q, 0 < |q| < 1.…”

On simultaneous approximation of the values of certain Mahler functions

“…In particular, s k ∈ {0, 1} and, moreover, the indexes k with s k = 1 form a so-called self-generating set, see [10]. In [1] Adamczewski proved that the numbers S(α) and S(α 4 ) are algebraically independent, if α, 0 < |α| < 1, is algebraic. Further, by using the gap properties of the series S(z) an upper bound [8].…”