Beta-expansion and continued fraction expansion over formal Laurent series

Let x I be an irrational element and n 1, where I is the unit disc in the field of formal Laurent series , we denote by kn(x) the number of exact partial quotients in continued fraction expansion of x, given by the first n digits in the β-expansion of x, both expansions are based on . We obtain that where Q*(x),Q*(x) are the upper and lower constants of x, respectively. Also, a central limit theorem and an iterated logarithm law for {kn(x)}n 1 are established.