NON CONVERGING CONTINUED FRACTIONS RELATED TO THE STERN DIATOMIC SEQUENCE

NON-CONVERGING CONTINUED FRACTIONS RELATED TO THE STERN DIATOMIC SEQUENCE by Boris Adamczewski 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 transcen- dence 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. 1. Introduction Given an integer a ≥ 2, it was recently observed in [7] that the regular continued fraction (1.1) C(a) = a+ 1 a2 + 1 a4 + 1 . . . + 1 a2n + 1 . . . , denoted in the sequel as usual by [a, a2, a4, . . . , a2 n , . . .], is transcendental. This is a con- sequence of Roth's theorem and follows directly from a result of Davenport and Roth [5] concerning the growth of denominators of convergents to an algebraic number. Quite surpris- ingly, the author of the present note was not able to pick up the scent of this simple example in the older literature though a function field analogue previously appeared in [12]. Indeed, viewed as a Laurent series in F2((1/x)), the continued fraction C(x) has the remarkable prop- erty of being a cubic element over the field F2(x).

• stern diatomic


• also related


• transcendental over


• following functional


• notice also


• main results


• functional equations


• rational functions


• results presented