Hankel determinants of sums of consecutive weighted Schröder numbers

Abstract

For a real number $t$, let $r_{\ell }(t)$ be the total weight of all $t$-large Schröder paths of length $\ell$, and $s_{\ell }(t)$ be the total weight of all $t$-small Schröder paths of length $\ell$. For constants $\alpha ,\beta$, in this article we derive recurrence formulae for the determinats of the Hankel matrices ${\det }_{1\le i,j\le n}(\alpha r_{i+j-2}(t)+\beta r_{i+j-1}(t))$, ${\det }_{1\le i,j\le n}(\alpha r_{i+j-1}(t)+\beta r_{i+j}(t))$, ${\det }_{1\le i,j\le n}(\alpha s_{i+j-2}(t)+\beta s_{i+j-1}(t))$, and ${\det }_{1\le i,j\le n}(\alpha s_{i+j-1}(t)+\beta s_{i+j}(t))$ combinatorially via suitable lattice path models.

15 pages