Continued fractions of certain Mahler functions

Abstract

We investigate the continued fraction expansion of the infinite product g(where the polynomial P (x) satisfies P (0) = 1 and deg(P ) < d. We construct relations between the partial quotients of g(x) which can be used to get recurrent formulae for them. We provide formulae for the cases d = 2 and d = 3. As an application, we prove that for P (x) = 1 + ux where u is an arbitrary rational number except 0 and 1, and for any integer b with |b| > 1 such that g(b) = 0 the irrationality exponent of g(b) equals two. In the case d = 3 we provide a partial analogue of the last result with several collections of polynomials P (x) giving the irrationality exponent of g(b) strictly bigger than two.

On the automaticity of sequences defined by the Thue–Morse and period-doubling Stieltjes continued fractions

“…On the other hand, continued fraction expansions of automatic numbers have been studied for calculating Hankel determinants and irrationality exponents [22,23,13,19,6]. In 1998, Allouche, Peyrière, Wen and Wen proved that all the Hankel determinants of the Thue-Morse sequence are nonzero [3].…”

“…The first terms of these sequences are listed below. 3,4,6,8,11,14,18,20,22,16,4, −32, −93, −220, . .…”

On the spectrum of irrationality exponents of Mahler numbers

“…Let a = (s 3 , −s 2 (s 2 + 1)). In this case we compute As for the collection (a), we need to check that Φ(g a ) does not contain any other primitive gap [u, v] with u < 7 which is obvious (by (24), we have the big gaps [2,5] and [6,13] in Φ(g a ).…”

“…However in many cases, as soon as we know that µ(f (b)) > 2 we can compute the precise value of the irrationality exponent of f (b) after computing only finitely many convergents of f . We demonstrate the method by computing the irrationality exponents of all Mahler numbers from [2] for which we know that µ(f (b)) > 2.…”

On the irrationality measure of the Thue–Morse constant

“…We will consider this result in more detail in the next subsection. Later, Badziahin [5] provided a continued fraction expansion for the functions of the form…”