Hankel determinants for convolution powers of Catalan numbers

Abstract

The Hankel determinants ${\left(\frac{r}{2(i+j)+r}\left(\genfrac{}{}{0ex}{}{2(i+j)+r}{i+j}\right)\right)}_{0\le i,j\le n-1}$ of the convolution powers of Catalan numbers were considered by Cigler and Krattenthaler. We evaluate these determinants for $r\le 31$ by finding shifted periodic continued fractions, which arose in application of Sulanke and Xin’s continued fraction method. These include some of the conjectures of Cigler as special cases. We also conjecture a polynomial characterization of these determinants. The same technique is used to evaluate the Hankel determinants ${\left(\left(\genfrac{}{}{0ex}{}{2(i+j)+r}{i+j}\right)\right)}_{0\le i,j\le n-1}$. Similar results are obtained.