# Continued fractions of certain Mahler functions

## TL;DR

This paper studies the continued fraction expansions of specific Mahler functions defined by infinite products, deriving recurrence relations for partial quotients, and analyzing their irrationality exponents at certain points.

## Contribution

It introduces new recurrence relations for partial quotients of Mahler functions and determines their irrationality exponents for particular polynomial cases.

## Key findings

- For $d=2$, irrationality exponent of $g(b)$ is exactly 2 for certain rational $u$ and integers $b$.
- For $d=3$, some polynomial choices yield irrationality exponents greater than 2.
- Constructed recurrence relations enable analysis of continued fractions of these 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.