Hankel determinants of sums of consecutive weighted Schröder numbers
Let \(\{a_l\}_{l\in \mathbb{Z}_+}\) be a sequence of numbers and let NEWLINE\[NEWLINEA_n^{(k)}=(a_{k+i+j-2})_{i,j=1}^nNEWLINE\]NEWLINE be the corresponding Hankel matrix. The determinants of such matrices were explicitly found in the case when \(a_l\) are Catalan numbers \(c_l\), Motzkin numbers \(m_l\), large and small Schöder numbers \(r_l\) and \(c_l\).NEWLINENEWLINEThese numbers count the number of lattice paths with fixed ends and some restrictions on these paths. In particular, the large Schöder numbers \(r_l\) count the number of lattice paths from \((0,0)\) to \((2l,0)\) using up steps \(U\), down steps \(D\) and level steps \(L=(2,0)\) that never pass below the \(x\)-axis. The small Schöder numbers \(s_l\) count large Schöder paths of length \(l\) without level steps on the \(x\)-axis.NEWLINENEWLINEThere exist generalizations of these numbers called weight versions of Catalan, Motzkin and Schöder numbers.NEWLINENEWLINEIn particular, the large Schöder numbers \(r^t_l\) and \(c^t_l\) are defined as follows. These steps \(U\), \(D\), \(L\) have weights 1, 1, t. Let \(r_l^t\) (and \(s_l^t\)) denote the total weight of all weighted large (small) Schöder paths of length \(l\).NEWLINENEWLINEThe main result of the paper is the following. The authors consider the determinantsNEWLINENEWLINENEWLINE\[NEWLINE\Theta_n=(1+t)^{-\binom{n}{2}}\det_{1\leq i,j\leq n}(\alpha r^t_{i+j-2}+\beta r^t_{i+j-1}),NEWLINE\]NEWLINENEWLINENEWLINENEWLINE\[NEWLINE \Phi_n=(1+t)^{-\binom{n+1}{2}}\det_{1\leq i,j\leq n}(\alpha r^t_{i+j-1}+\beta r^t_{i+j}),NEWLINE\]NEWLINENEWLINENEWLINENEWLINE\[NEWLINE\Psi_n=(1+t)^{-\binom{n}{2}}\det_{1\leq i,j\leq n}(\alpha s^t_{i+j-2}+\beta s^t_{i+j-1}),NEWLINE\]NEWLINENEWLINENEWLINENEWLINE\[NEWLINE\Gamma_n= (1+t)^{-\binom{n+1}{2}}\det_{1\leq i,j\leq n}(\alpha s^t_{i+j-1}+\beta s^t_{i+j}).NEWLINE\]NEWLINENEWLINENEWLINEHere \(\alpha\), \(\beta\) are some parameters.NEWLINENEWLINEThe authors find generating functions of \(\Theta_n,\Phi_n,\Psi_n,\Gamma_n\) and explicit formulas for these quantities.
scientific article; zbMATH DE number 6078349