## Illinois Journal of Mathematics

### Über summen von Rudin-Shapiroschen koeffizienten

#### Abstract

The Rudin-Shapiro coefficients $\{a(n)\}$ are an infinite sequence of $\pm 1$'s, defined recursively by $a(0)=1$, $a(2n)=a(n)$, and $a(2n + 1)=(-1)^{n}a(n)$, $n \geq 0$ Various formulas are developed for the $n$th partial sum $s(n)$ and the $n$th alternating partial sum $t(n)$ of this sequence. These formulas are then used to show that $\sqrt{3/5} < s(n)/\surd n < \surd 6$ and $0 \leq; t(n)/\surd n < \surd 3$, $n \geq 1$ where the inequalities are sharp and the ratios are dense in the two intervals. For a given $n \geq 1$, the equation $s(k)= n$ is shown to have exactly $n$ solutions $k$.

#### Article information

Source
Illinois J. Math., Volume 22, Issue 1 (1978), 126-148.

Dates
First available in Project Euclid: 20 October 2009

https://projecteuclid.org/euclid.ijm/1256048841

Digital Object Identifier
doi:10.1215/ijm/1256048841

Mathematical Reviews number (MathSciNet)
MR0476686

Zentralblatt MATH identifier
0371.10009

Subjects
Primary: 10L10

#### Citation

Brillhart, John; Morton, Patrick. Über summen von Rudin-Shapiroschen koeffizienten. Illinois J. Math. 22 (1978), no. 1, 126--148. doi:10.1215/ijm/1256048841. https://projecteuclid.org/euclid.ijm/1256048841