Continued fractions of certain Mahler functions

D. Badziahin

Abstract

We investigate the continued fraction expansion of the infinite products \(g(x) = x^{-1}\prod_{t=0}^\infty P(x^{-d^t})\) where polynomials \(P(x)\) satisfy \(P(0)=1\) and \(\deg(P)< d\). We construct relations between partial quotients of \(g(x)\) which can be used to get recurrent formulae for them. We provide that 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 \(g(b)\neq0\) 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.

Keywords: Mahler function, Mahler number, irrationality exponent, continued fraction of Laurent series, Pade approximation.

AMS Subject Classification: Primary 11B83; secondary 11J82, 41A21.