Lattice Paths Between Diagonal Boundaries

A bivariate symmetric backwards recursion is of the form $d[m,n]=w_{0}(d[m-1,n]+d[m,n-1])+\omega_{1}(d[m-r_{1},n-s_{1}]+d[m-s_{1},n-r_{1}])+\dots+\omega_{k}(d[m-r_{k},n-s_{k}]+d[m-s_{k},n-r_{k}])$ where $\omega_{0},\dots\omega_{k}$ are weights, $r_{1},\dots r_{k}$ and $s_{1},\dots s_{k}$ are positive integers. We prove three theorems about solving symmetric backwards recursions restricted to the diagonal band $x+u