Minors of a class of Riordan arrays related to weighted partial Motzkin paths

A partial Motzkin path is a path from (0,0)(0,0) to (n,k)(n,k) in the XOYXOY-plane that does not go below the XX-axis and consists of up steps U=(1,1)U=(1,1), down steps D=(1,−1)D=(1,−1) and horizontal steps H=(1,0)H=(1,0). A weighted partial Motzkin path is a partial Motzkin path with the weight assignment that all up steps and down steps are weighted by 11, the horizontal steps are endowed with a weight xx if they are lying on XX-axis, and endowed with a weight yy if they are not lying on XX-axis. Denote by Mn,k(x,y)Mn,k(x,y) the weight function of all weighted partial Motzkin paths from (0,0)(0,0) to (n,k)(n,k), and M=(Mn,k(x,y))n≥k≥0M=(Mn,k(x,y))n≥k≥0 the infinite lower triangular matrices. In this paper, we consider the sums of minors of second order of the matrix MM, and obtain a lot of interesting determinant identities related to MM, which are proved by bijections using weighted partial Motzkin paths. When the weight parameters (x,y)(x,y) are specialized, several new identities are obtained related to some classical sequences involving Catalan numbers. Besides, in the alternating cases we also give some new explicit formulas for Catalan numbers.