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 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 conjectured 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.

29 pages

References in corpus (4)

• Hankel determinant solutions to several discrete integrable systems and the Laurent Property
• Some nice Hankel determinants
• Some conjectures on the ratio of Hankel transforms for sequences and series reversion
• Catalan numbers, Hankel determinants and Fibonacci polynomials