Counting humps and peaks in generalized Motzkin paths

Abstract Let us call a lattice path in ℤ × 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^HiD, i≥0. This generalizes recent results concerning the cases when k=1 and a=1 or a=∞. A similar relation may be given involving peaks (consecutive steps of the form UD).