Stieltjes continued fractions related to the paperfolding sequence and Rudin-Shapiro sequence

Abstract

We investigate two Stieltjes continued fractions given by the paperfolding sequence and the Rudin-Shapiro sequence. By explicitly describing certain subsequences of the convergents Pn(x)/Qn(x) modulo 4, we give the formal power series expansions (modulo 4) of these two continued fractions and prove that they are congruent modulo 4 to algebraic series in Z[[x]]. Therefore, the coefficient sequences of the formal power series expansions are 2-automatic. Write Qn(x)=∑i≥0an,ixi. Then (Qn(x))n≥0 defines a two-dimensional coefficient sequence (an,i)n,i≥0. We prove that the coefficient sequences (an,imod4)n≥0 introduced by both (Qn(x))n≥0 and (Pn(x))n≥0 are 2-automatic for all i≥0. Moreover, the pictures of these two dimensional coefficient sequences modulo 4 present a kind of self-similar phenomenon.