Counting humps and peaks in generalized Motzkin paths

Let us call a lattice path in Z×ZZ×Z from (0,0)(0,0) to (n,0)(n,0) using U=(1,k)U=(1,k), D=(1,−1)D=(1,−1), and H=(a,0)H=(a,0) steps and never going below the xx-axis, a (k,a)(k,a)-path   (of order nn). A super     (k,a)(k,a)-path   is a (k,a)(k,a)-path which is permitted to go below the xx-axis. We relate the total number of humps in all of the (k,a)(k,a)-paths of order nn to the number of super (k,a)(k,a)-paths, where a hump is defined to be a sequence of steps of the form UHiDUHiD, i≥0i≥0. This generalizes recent results concerning the cases when k=1k=1 and a=1a=1 or a=∞a=∞. A similar relation may be given involving peaks   (consecutive steps of the form UDUD).