## Continued fractions of certain Mahler functions

#### 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.

: Primary 11B83; secondary 11J82, 41A21.

This paper is available as a pdf (192kB) file. It is also on the arXiv: arxiv.org/abs/1702.07457.

 Sunday, July 23, 2017