Counting humps and peaks in generalized Motzkin paths

Let us call a lattice path in $\mathbb{Z}\times \mathbb{Z}$ from $(0,0)$ to $(n,0)$ using $U=(1,k)$, $D=(1,?1)$, and $H=(a,0)$ steps and never going below the $x$-axis, a $(k,a)$-path (of order $n$). A super   $(k,a)$-path is a $(k,a)$-path which is permitted to go below the $x$-axis. We relate the total number of humps in all of the $(k,a)$-paths of order $n$ to the number of super $(k,a)$-paths, where a hump is defined to be a sequence of steps of the form $U{H}^{i}D$, $i\ge 0$. This generalizes recent results concerning the cases when $k=1$ and $a=1$ or $a=\infty$. A similar relation may be given involving peaks (consecutive steps of the form $UD$).